Math 100G/L Introduction to Algebra and Finance BYU-Idaho
MESSAGE FROM THE FIRST PRESIDENCY Dear Brothers and Sisters: Latter-day Saints have been counseled for many years to prepare for adversity by having a little money set aside. Doing so adds immeasurably to security and well-being. Every family has a responsibility to provide for its own needs to the extent possible. We encourage you wherever you may live in the world to prepare for adversity by looking to the condition of your finances. We urge you to be modest in your expenditures; discipline yourselves in your purchases to avoid debt. Pay off debt as quickly as you can, and free yourselves from this bondage. Save a little money regularly to gradually build a financial reserve. If you have paid your debts and have a financial reserve, even though it be small, you and your family will feel more secure and enjoy greater peace in your hearts. May the Lord bless you in your family financial effort. The First Presidency
(From the pamphlet ALL IS SAFELY GATHERED IN: FAMILY FINANCES published by the Church.)
Table of Contents Chapter 1 Arithmetic……….…………………………………………………………...1 Section 1.1………………………………………………………………………………...2 Addition and Multiplication Facts from 1+1 to 15 × 15
Section 1.2……………………………………………………………………………..…10 Rounding and Estimation; Life Plan
Section 1.3………………………………………………………………………………..15 Add, Subtract, Multiply, Divide Decimals; Income and Expense
Section 1.4……………………………………………………………………………..…34 Add, Subtract, Multiply, Divide Fractions; Unit Conversions
Chapter 2 Calculators and Formulas ……………………………………………49 Section 2.1…………………………………………………………………………..……50 Exponents Introduction, Order of Operations, Calculator Usage
Section 2.2……………………………………………………..…………………………61 Variables and Formulas
Section 2.3……………………………………………………..…………………………76 Formulas and Spreadsheet Usage
Chapter 3 Algebra.………………....……………………………………………………89 Section 3.1……………………………………………………………………………….90 Linear Equations and Applications
Section 3.2………………………………………………………………………………108 Linear Equations with Fractions; Percent Applications
Section 3.3………………………………………………………………………………121 Exponents Revisited; Loan Payment and Savings Equations
Chapter 4 Graphs and Charts…………………………………..………………...137 Section 4.1………………………………………………………………………………138 Maps and Coordinate Graphs
Section 4.2………………………………………………………………………………148 Graphing Lines and Finding Slope
Section 4.3………………………………………………………………………………162 Using Slope and Writing Equations of Lines
1
Chapter 1: ARITHMETIC
Overview Arithmetic
1.1 Facts 1.2 Rounding and Estimation 1.3 Decimals 1.4 Fractions
2 2
Section 1.1
Everyone has to start somewhere, and that start, for you, is right here. When you first started learning math, you probably learned the names for numbers, and then you started to add: 3apples + 7apples equals how many apples? Well 10, of course. Facts
My guess is that you caught on to what you were doing and can now add M&M’s, coconuts, gallons of water, money etc. From the beginning I am going to assume you know how to add in your head up to 15+15. If you don’t, please make up some flash cards and get those in your It is similar learning the alphabet before learning to read. We need the addition facts brain. to be available for to instant recall. Soon after addition was learned, I bet someone told you that there was a shortcut when you had to add some numbers over and over. For example: 3+3+3+3+3+3+3 = 21 7 If you notice, there are seven 3’s. 3, seven times, turns out to be 21, so we write it as 7×3 = 21. One of the best coincidences of the world is that 7, three times, is also 21. 3×7 = 21 Such a switching works for any numbers we pick: 4×5 = 20 and 5×4 = 20 3×13 = 39 and 13×3 = 39 Since we will be using the multiplication facts almost as much as we will be using the addition facts, you need to also memorize the multiplication facts up to 15×15. Learn them well, and you will be able to catch on to everything else quite nicely.
Section 1.1
3
Section 1.1 Exercises Part A 1. Make flash cards up to 15+15 and 15×15. 2. Memorize the addition and multiplication facts up to 15+15 and 15×15. 3. Fill out the Addition/Subtraction Monster. Time yourself. Write the time it takes on the paper. Correct the Addition/Subtraction Monster using your flashcards. 4. Fill out the Multiplication Monster. Time yourself. Write the time on the paper. Correct the Addition/Subtraction Monster using your flashcards.
Assignment 1.1a
4
Addition/Subtraction Monster
Name __________________
12 + 13
5 +6
5 + 10
12 − 9
5 +9
8 + 11
5 + 11
14 4
6+ 6
7 + 12
15 8
10 + 10
10 7
6 + 11
6 + 12
6 + 13
7+ 7
14 7
7 +9
9 + 13
6 + 14
15 5
11 + 11
7 5
12 − 4
10 + 12
8 + 10
13 − 8
5 +5
8 + 13
5 + 12
7 +8
9+ 9
5 + 15
9 + 11
9 + 12
15 − 6
13 − 5
9 + 15
8 + 15
6+ 7
13 − 9
8 + 12
10 + 13
10 + 14
10 + 15
7 + 13
11 + 13
5+ 7
11 + 12
14 − 9
11 + 14
11 + 15
8 +9
10 − 6
8−7
12 + 12
6 + 10
12 + 14
8 +8
12 − 7
12 − 8
14 + 14
12 − 6
9− 7
13 + 14
10 − 5
7 + 14
6 +9
13 − 7
13 − 6
9 + 10
6+ 8
14 + 15
14 − 10
12 + 15
14 − 8
8 + 14
14 − 6
10 + 11
8− 5
15 − 11
15 − 10
15 − 9
9 −8
7 + 10
9 + 14
13 + 15
7 + 11
5 + 14
6 + 15
15 − 7
5 + 13
7 + 15
5 +8
7−6
13 + 13
8 −6
9 −5
9 −6
15 − 4
15 + 15
13 − 4
14 − 5
Time_________ Assignment 1.1 a
5
Multiplication Monster
Name __________________
12×13=
5×6=
5×10=
12×9=
5×9=
8×11=
5×11=
14×4=
6×6=
7×12=
15×8=
10×10=
10×7=
6×11=
6×12=
6×13=
7×7=
14×7=
7×9=
9×13=
6×14=
15×5=
11×11=
7×5=
12×4=
10×12=
8×10=
13×8=
5×5=
8×13=
5×12=
7×8=
9×9=
5×15=
9×11=
9×12=
15×6=
13×5=
9×15=
8×15=
6×7=
13×9=
8×12=
10×13=
10×14=
10×15
7×13=
11×13=
5×7=
11×12=
14×9=
11×14=
11×15=
8×9=
10×6=
8×7=
12×12=
6×10=
12×14=
8×8=
12×7=
12×8=
14×14=
12×6=
9×7=
13×14=
10×5=
7×14=
6×9=
13×7=
13×6=
9×10=
6×8=
14×15=
14×10=
12×15=
14×8=
8×14=
14×6=
10×11=
8×5=
15×11=
15×10=
15×9=
9×8=
7×10=
9×14=
13×15=
7×11=
5×14=
6×15=
15×7=
5×13=
7×15=
5×8=
7×6=
13×13=
8×6=
9×5=
9×6=
15×4=
15×15=
13×4=
14×5=
Time_________
Assignment 1.1a
6
Section 1.1 Exercises Part B
Addition/Subtraction Monster 2 9−6
12− 4
5 + 10
6 + 15
15 − 5
8 + 11
12 − 9
14 − 4
6 +6
9−7
15 − 8
10 + 10
10 − 7
6 + 11
13 − 7
5 +8
7 +7
7 + 12
15 − 10
9 + 13
6 + 14
12 + 13
7 −5
13 + 15
5 + 11
10 + 12
8 + 10
15 − 7
14 − 7
8 + 13
5 + 12
7 +8
9 +9
5 + 15
9 + 11
9 + 12
6 + 13
5 +5
9 + 15
8 + 15
6 +7
11 + 15
8 + 12
13 − 5
10 + 14
10 + 15
7 + 13
11 + 13
5 +7
11 + 12
11 + 11
11 + 14
13 − 8
8 +9
10 − 6
5 +9
12 + 12
14 − 9
12 + 14
8 +8
12 − 7
10 + 13
14 + 14
12 − 6
15 + 15
13 + 14
10 − 5
7 + 14
12 − 8
6 +8
13 − 6
9 + 10
5 +6
14 + 15
6 + 10
12 + 15
14 − 8
8 + 14
14 − 6
10 + 11
8 −5
15 − 11
13 − 9
15 − 9
6 +9
7 + 10
9 + 14
7 −6
7 + 11
5 + 14
15 − 6
6 + 12
14 − 10
7 + 15
9 −8
7 +9
13 + 13
8−6
9 −5
5 + 13
15 − 4
8 −7
13 − 4
14 − 5
Time_________ Assignment 1.1b
7
Multiplication Monster 2 9×6=
12×4=
5×10=
6×15=
15×5=
8×11=
12×9=
14×4=
6×6=
9×7=
15×8=
10×10=
10×7=
6×11=
13×7=
5×8=
7×7=
7×12=
15×10=
9×13=
6×14=
12×13=
7×5=
13×15=
5×11=
10×12=
8×10=
15×7=
14×7=
8×13=
5×12=
7×8=
9×9=
5×15=
9×11=
9×12=
6×13=
5×5=
9×15=
8×15=
6×7=
11×15=
8×12=
13×5=
10×14=
10×15=
7×13=
11×13=
5×7=
11×12=
11×11=
11×14=
13×8=
8×9=
10×6=
5×9=
12×12=
14×9=
12×14=
8×8=
12×7=
10×13=
14×14=
12×6=
15×15=
13×14=
10×5=
7×14=
12×8=
6×8=
13×6=
9×10=
5×6=
14×15=
6×10=
12×15=
14×8=
8×14=
14×6=
10×11=
8×5=
15×11=
13×9=
15×9=
6×9=
7×10=
9×14=
7×6=
7×11=
5×14=
15×6=
6×12=
14×10=
7×15=
9×8=
7×9=
13×13=
8×6=
9×5=
5×13=
15×4=
8×7=
13×4=
14×5=
Time_________ Assignment 1.1b
8
Section 1.1 Exercises Part C
Addition/Subtraction Monster
Name __________________
12 + 13
5 +6
5 + 10
12 − 9
5 +9
8 + 11
5 + 11
14 − 4
6+ 6
7 + 12
15 − 8
10 + 10
10 − 7
6 + 11
6 + 12
6 + 13
7+ 7
14 − 7
7 +9
9 + 13
6 + 14
15 − 5
11 + 11
7−5
12 − 4
10 + 12
8 + 10
13 − 8
5 +5
8 + 13
5 + 12
7+8
9+ 9
5 + 15
9 + 11
9 + 12
15 − 6
13 − 5
9 + 15
8 + 15
6+ 7
13 − 9
8 + 12
10 + 13
10 + 14
10 + 15
7 + 13
11 + 13
5+ 7
11 + 12
14 − 9
11 + 14
11 + 15
8 +9
10 − 6
8−7
12 + 12
6 + 10
12 + 14
8 +8
12 − 7
12 − 8
14 + 14
12 − 6
9− 7
13 + 14
10 − 5
7 + 14
6 +9
13 − 7
13 − 6
9 + 10
6+ 8
14 + 15
14 − 10
12 + 15
14 − 8
8 + 14
14 − 6
10 + 11
8− 5
15 − 11
15 − 10
15 − 9
9 −8
7 + 10
9 + 14
13 + 15
7 + 11
5 + 14
6 + 15
15 − 7
5 + 13
7 + 15
5 +8
7−6
13 + 13
8 −6
9 −5
9 −6
15 − 4
15 + 15
13 − 4
14 − 5
Time_________ Assignment 1.1 c
9
Multiplication Monster
Name __________________
12×13=
5×6=
5×10=
12×9=
5×9=
8×11=
5×11=
14×4=
6×6=
7×12=
15×8=
10×10=
10×7=
6×11=
6×12=
6×13=
7×7=
14×7=
7×9=
9×13=
6×14=
15×5=
11×11=
7×5=
12×4=
10×12=
8×10=
13×8=
5×5=
8×13=
5×12=
7×8=
9×9=
5×15=
9×11=
9×12=
15×6=
13×5=
9×15=
8×15=
6×7=
13×9=
8×12=
10×13=
10×14=
10×15
7×13=
11×13=
5×7=
11×12=
14×9=
11×14=
11×15=
8×9=
10×6=
8×7=
12×12=
6×10=
12×14=
8×8=
12×7=
12×8=
14×14=
12×6=
9×7=
13×14=
10×5=
7×14=
6×9=
13×7=
13×6=
9×10=
6×8=
14×15=
14×10=
12×15=
14×8=
8×14=
14×6=
10×11=
8×5=
15×11=
15×10=
15×9=
9×8=
7×10=
9×14=
13×15=
7×11=
5×14=
6×15=
15×7=
5×13=
7×15=
5×8=
7×6=
13×13=
8×6=
9×5=
9×6=
15×4=
15×15=
13×4=
14×5=
Time_________
Assignment 1.1 c
10
Now, you know that some arithmetic problems may get long and tedious, so you can understand why some folks choose to estimate Rounding and and round numbers. Rounding is the quickest, so we will tackle that Estimation first. In rounding, we decide to not keep the exact number that someone gave us. For example:
Section 1.2
Rounding If I have $528.37 in the bank, I might easily say that I have about $500. I have just rounded to the nearest hundred. On the other hand, I might be a little more specific and say that I have about (still not exact) $530. I have just rounded to the nearest ten. Here are the places: Just to make sure you are clear on it, here is a big example:
s n o lil i B
s n io ll i M ed r d n u H
s n ilo il M n e T
s d n sa u o h T s d n er io d n lli u M H
s d asn u o h T n e T
s d an s u o h T
s ed r d n u H
s n e T
es n O
s h t n e T
s h t ed r d n u H
s h t d an s u o h T
s h t d asn u o h T n e T
s h t d n sa u o h T ed r d n u H
s h t n io lli M
6,731,239,465.726409 Example: Round to the nearest hundredth: 538.4691 This number is right between 538.46 and 538.47 Which one is nearest? The 9 tells us that we are closer to 538.47 2nd Example: Round to the nearest thousand: 783,299.4321 This number is right between 783,000 and 784,000 Which one is nearest? The 2 in the hundreds tells us that we are closer to : 783,000
Section 1.2
11
LAST EXAMPLE Round $4,278.23 to the nearest hundred $4,300.00 Decide if our number is closer to the nearest $4,278.23 hundred above the number or below the number $4,200.00 Change our number to the one it is closer to $4,278.23 ≈ $4,300.00 Answer: $4,300.00
Estimation Estimation 1. Round to the highest value. 2. Do the easy problem.
Once rounding is understood, it can be used as a great tool to make sure that we have not missed something major in our computations. If we have a problem like: 3,427,000 × 87.3 We could see about where the answer is if we estimate first: Round each number to the greatest value you can 3,000,000 × 90 Voila! Our answer will be about 270,000,000 We should note that the real answer is: 299,177,100 but the estimation will let us know that we are in the right ball park. It ensures that our answer makes sense.
LAST EXAMPLE Multiply by rounding: 986.7 4.9 Round the numbers 986.7 ≈ 1,000 4.9 ≈ 5 Multiply the rounded numbers together 1,000 5 5,000 Our answer for 986.7 4.9 will be about 5,000 986.7 4.9 ≈ 5,000
Section 1.2
12
Section 1.2 Exercises Part A 1. Round 3,254.07 to the nearest ten. 2. Round 2,892.56 to the nearest tenth. 3. Round 39,454 to the nearest ten thousand. 4. Round 189 to the nearest ten. 5. Round 3,250.07 to the nearest tenth. 6. Round 2,892.56 to the nearest hundred. 7. Round 39, 454 to the nearest ten. 8. Round 189 to the nearest hundred.
Estimate the following. 9.
21 × 3250.07
10. 138.9 × 2892
11. 42 × 189
12.
369.456
13. 58 × 39
14. 351 × 44
3.987
Preparation: 15. Find the monthly income for 5 different jobs and be ready to share them with your group.
Answers: 3,250 1. 2,892.6 2. 40,000 3. 190 4. 3,250.1 5. 2,900 6. 39,450 7. 200 8.
9. 10. 11. 12. 13. 14. 15.
About 60,000 About 300,000 About 8,000 About 100 About 2,400 About 16,000 Discuss it together
Assignment 1.2a
13
Section 1.2 Exercises Part B 1. 2. 3. 4. 5. 6.
Round 7,254.07 to the nearest ten. Round 2,862.843 to the nearest hundredth. Round 538,484 to the nearest ten thousand. Round 189.59 to the nearest ten. Round 3,250.647 to the nearest tenth. Round 2,892.56385 to the nearest thousandth.
7. Round 34,454 to the nearest thousand. 8. Round 189,364,529.83 to the nearest million. 9. Describe what possible problems students could have with rounding.
Estimate the following. 10.
51 × 3250.07
11.
438.9 × 2,892.07
12.
32 × 789
13.
569.456
14.
58 × 391
15.
54,200
6.1987
12
16. Working with your group, find the yearly income for 10 of the jobs brought in by group members.
17. As a group, estimate a monthly budget for a family with a few children living in your area. Please include estimates of costs for housing, transportation, food, utilities, and clothing. 18. Enter the budget into a spreadsheet document.
Answers: 7,250 1. 2,862.84 2. 540,000 3. 190 4. 3,250.6 5. 2,892.564 6. 34,000 7. 189,000,000 8. 9.
d vs. dth, lack of 1th, any others
10. 11. 12. 13. 14. 15. 16. 17.
About 150,000 About 1,200,000 About 24,000 About 100 About 24,000 About 5,000 Make sure they are all there. Should look neat.
18.
Complete when everyone can do it.
Assignment 1.2b
14
Section 1.2 Exercises Part C 1. 2. 3. 4. 5. 6. 7.
Round 7,254.07 to the nearest tenth. Round 2,862.843 to the nearest ten. Round 538,484 to the nearest thousand. Round 139.79 to the nearest ten. Round 3,250.647 to the nearest hundredth. Round 2,892.56385 to the nearest thousand. Round 34,454 to the nearest thousand.
8. Round 189,364,529.83 to the nearest ten million. Estimate the following. 9.
41 × 7250.07
10.
43 × 9.07
11.
82 × 2,890
12.
639.456
13.
58 × 391.04
14.
56,200
6.1987
12
Begin “Life Plan” Portfolio Project. 15. Imagine your life five years from now. Estimateone month of what you think your expenses and income will be at that time.. 16. Create your own spreadsheet document to record your one month estimated expenses and income. Remember, you are forecasting five years into the future and recording one a month estimate of your anticipated income and expenses into a spreadsheet. Prepare for “Budget and Expenses” Portfolio Project. 17. Report to your group that you have started keeping track of your income and expenses. 18. Receive reports from your group members that they have started tracking their current income and expenses.
Answers: 7,254.1 1. 2,860 2. 538,000 3. 140 4. 3,250.65 5. 3,000 6. 34,000 7.
10. 11. 12. 13. 14. 15. 16.
About 360 About 240,000 About 100 About 24,000 About 6,000 Include any expenses you can think of. Save it as “Life Plan”. You will submit it to your teacher in this lesson.
8.
190,000,000
17.
9.
About 280,000
18.
Start your record, then report to your progress to your group by email, phone, letter, carrier pigeon… Complete when everyone has done it.
15
Section 1.3
DEFINITIONS & BASICS
Decimals
1) Like things – In addition and subtraction we must only deal with like things. Example: If someone asks you 5 sheep + 2 sheep = you would be able to tell them 7 sheep. What if they asked you 5 sheep + 2 penguins = We really can’t add them together, because they aren’t like things.
2) We do not need like things for multiplication and division. 3) Negative – The negative sign means “opposite direction.” Example: −5.3 is just 5.3 in the opposite direction
−5.3
0
5.3
Example : − is just in the opposite direction. Example: −7 5
−12 , because they are both headed in that direction
4) Decimal – Deci is a prefix meaning 10. Since every place value is either 10 times larger or smaller than the place next to it, we call each place a decimal place. 5) Place Values – Every place on the left or right of the decimal holds a certain value
Arithmetic of Decimals, Positives and Negatives LAWS & PROCESSES
Addition of Decimals 1. Line up decimals 2. Add in columns 3. Carry by 10’s Section 1.3
16
EXAMPLE Add. 3561.5 + 274.38 3561.5 + 274.38
+
1. Line up decimals
3 5 6 1. 5 2 7 4. 3 8 5. 8 8
2. Add in columns
1
3 5 6 1. 5 + 2 7 4. 3 8 3 8 3 5. 8 8
3. Carry by 10’s. Carry the 1 and leave the 3.
Subtraction of Decimals 1. 2. 3. 4.
Biggest on top Line up decimals; subtract in columns. Borrow by 10’s Strongest wins. EXAMPLE Subtract. 283.5 – 3,476.91 - 3476.91 283.5 - 3 4 7 6. 9 1 2 8 3. 5 3. 4 1
1.Biggest on top
2. Line up decimals; subtract in columns
3
- 3 4 17 6. 9 1 2 8 3. 5 3 1 9 3. 4 1
3. Borrow by 10’s. Carry the 1 and leave the 3.
- 3 34 17 6. 9 1 2 8 3. 5 - 3 1 9 3. 4 1
3. Biggest one wins.
Section 1.3
17
Multiplication of Decimals Multiplication of Decimals 1. Multiply each place value 2. Carry by 10’s 3. Add 4. Right size.
1. Add up zeros or decimals 2. Negatives
EXAMPLES Start: 7 5 31
29,742 × 538 3. Add the pieces together.
237,936 892,260 +14,871,000 16,001,196
29,742 × 8 237,936
Next: 22 1
29,742 × 30 892,260 Last: 4321
29,742 × 500 14,871,000
Section 1.3
18
Start:
Final example with decimals:
-7414.3 × 9.46 444858 2965720 +66728700 -70139278
3. Add the pieces together.
2
21
74143 × 6 444858
Next: 1 11 74143 × 40 2965720 Last: 3 1 32
4. Right size. Total number of decimal places = 3. Answer is negative.
74143 × 900 66728700
The only thing left is to count the number of decimal places. We have one in the first number and two in the second. Final answer: -70139.278
Division of Decimals Division of Decimals 1. Move decimals 1. Set up. 2. Add zeros 2. Divide into first. 3. Multiply. 4. Subtract. 5. Drop down. 6. Write answer.
1. Remainder 2. Decimal
Section 1.3
19
EXAMPLES Step 1. No decimals to set up. Go to Step 2. Step 2.We know that 8 goes into 42 about 5 times. Step 3. Multiply 5×8
5 8 429 5 8 429
Step 4.subtract.
-40 53 8 429
Step 5. Bring down the 9 to continue on. Repeat steps 2-5
-40 29 53
Step 2: 8 goes into 29 about 3 times. Step 3: Multiply 3×8
8 429 -40 29 -24 5
Step 4: subtract. 8 doesn’t go into 5 (remainder)
Which means that 429 ÷ 8 = 53 R 5 or in other words 429 ÷ 8 = 53 85
Example: 5875 ÷ 22 2 22 5875 44 2 22 5875 -44 147 27 22 5875 -44 147 154
Step 2: 22 goes into 58 about 2 times. Step 3: Multiply 2×22 = 44 Step 4: Subtract. Step 5: Bring down the next column 22 goes into 147 about ???? times. Let’s estimate. 2 goes into 14 about 7 times – try that. Multiply 22×7 = 154 Oops, a little too big
Section 1.3
20
26 22 5875 -44 147 -132 155 267 22 5875 -44
Since 7 was a little too big, try 6. Multiply 6×22 = 132 Subtract. Bring down the next column. 22 goes into 155 about ????? times. Estimate. 2 goes into 15 about 7 times. Try 7
147 Multiply 22×7 = 154. It worked. -132 155 Subtract. -154 Remainder 1 1 5875 ÷ 22 = 267 R 1 or 267 221
An example resulting in a decimal: 4 Write as a decimal: 9 Step 1: Set it up. Write a few zeros, just to be 9 4.0000 safe. .4 Step 2: Divide into first. 9 goes into 40 about 4 times. 9 4.0000 Step 3. Multiply 4×9 = 36 -36 4 .44 9 4.0000 -36 40 -36 4 .444 9 4.0000 -36 40 -36 40 -36
Repeating decimal
Step 4. Subtract. Step 5. Bring down the next column. Repeat steps 2-4 Step 2: 9 goes into 40 about 4 times. Step 3: Multiply 4×9 = 36 Step 4: Subtract. Step 5. Bring down the next column. Repeat steps 2-4 Step 2: 9 goes into 40 about 4 times. Step 3: Multiply 4×9 = 36 Step 4: Subtract. This could go on forever!
44 Thus = .44444. . . which we simply write by .4 9 The bar signifies numbers or patterns that repeat. Section 1.3
21
Two final examples: -(.005)
358.4 .005 358.4
296 Step 1. Set it up and move the decimals
5 358400 7 5 358400 35 7 5 358400 -35 08 71 5 358400 -35 08 - 5 34 716 5 358400 -35 08 - 5 34 -30 40 7168 5 358400 -35 08 - 5 34 -30 40 -40 00
3.1 3.1 296 31 2960 .00
Step 2. Divide into first Step 3. Multiply down
Step 4. Subtract Step 5. Bring down Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step 5. Bring down Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step 5. Bring down
Repeat steps 2-5 as necessary Step 2: Divide into first
Step 3: Multiply down Step 4: Subtract Step 5: Bring down
9 31 2960 .00 279 9 31 2960 .00 -279 170 95. 31 2960 .00 -279 170 -155 150 95.4 31 2960 .00 -279 170 -155 150 -124 26 95.48 31 2960.000 -279 170 -155 150 -124 260 - 248 120
Section 1.3
22
71680 5 358400 -35 08 - 5 34 -30 40 -40 00 -0 0 -71,680
Repeat steps 2-5 as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract
31 2960.000
Step 6: Write answer
95.483 . . .
One negative in the srcinal problem gives a negative answer.
95.483 -279 170 -155 150 -124 260 - 248 120 -93 27
The decimal obviously keeps going. Round after a couple of decimal places.
COMMON MISTAKES
Two negatives make a positive - True in Multiplication and Division – Since a negative sign simply means opposite direction, when we switch direction twice, we are headed back the way we started. Example: -(-5) = 5 Example: -(-2)(-1)(-3)(-5) = - - - - -30 = -30 Example: -(-40 -8) = -(- -5) = -5
- False in Addition and Subtraction – With addition and subtraction negatives and positives work against each other in a sort of tug ‘o war. Whichever one is stronger will win.
Example: Debt is negative and income is positive. If there is more debt than income, then the net result is debt. If we are $77 in debt and get income of $66 then we have a net debt of $11 -77 + 66 = -11 On the other hand if we have $77 dollars of income and $66 of debt, then the net is a positive $11 77 – 66 = 11 Section 1.3
23
Example: Falling is negative and rising is positive. An airplane rises 307 feet and then falls 23 feet, then the result is a rise of 284 feet: 307 – 23 = 284 If, however, the airplane falls 307 feet and then rises 23 feet, then the result is a fall of 284 feet: -307 + 23 = -284 is negative andWhichever markup or is sales tax is positive. Other examples: Warmer is positiveDiscount and colder is negative. greater will give you the sign of the net result.
1) Percent: Percent can be broken up into two words: “per” and “cent” meaning per hundred, or in other words, hundredths. 7 31 53 = .07 = 7% = .31 = 31% = .53 = 53% 100 100 100 Notice the shortcut from decimal to percents : move the decimal right two places. Example:
LAWS & PROCESSES Converting Percents
Percents 1. If fraction, solve for decimals. for decimal to % 2. Move decimal 2 places. 1.2. Right Left for % to decimal 3. “OF” means times.
EXAMPLES .25=
25%
Convert .25 to a percent Move the decimal two places to the right because we are turning this into a percent .25=25% Section 1.3
24
What is
5
32
.15625
.15625=15.625% 5 32
as a percent? Turn the fraction into a decimal by dividing Move the decimal two places to the right because we are turning this into a percent
15.652%
Convert 124% to decimals 124%=1.24
Move decimal to the left because we arethe turning thistwo intoplaces a decimal 124%=1.24
Solving “Of” with Percents The most important thing that you should know about percents is that they never stand alone. If I were to call out that I owned 35%, the immediate response is, “35% of what?” Percents always are a percent of something. For example, sales tax is about 6% or 7% of your purchase. Since this is so common, we need to know how to calculate this. If you buy $25 worth of food and the sales tax is 7%, then the actual tax is 7% of $25. .07×$25 = $1.75 In math terms the word “of” means multiply.
EXAMPLES
25%=.25 . 25
64
16
What is 25% of 64? Turn the percent into a decimal Multiply the two numbers together 25% of 64 is 16
What is 13% of $25? 13%=.13
. 13 25
Turn the percent into a decimal
3.25
Multiply the two numbers together 13% of $25 is $3.25 What is 30% of 90 feet?
30%=.30
. 30
90
27
Turn the percent into a decimal Multiply the two numbers together 30% of 90 feet is 27 feet Section 1.3
25
Section 1.3 Exercises Part A Add. 1.
36,451 + 2,197
2.
143.29 + .923
3.
5,834,906.2 + 54.3227
Subtract. 4.
7- (-2) =
5.
-7 – 2 =
6.
-13 –(-10) =
7.
-18 + 5 =
8.
10 – 57 =
9.
-14 – 8 =
10.
234 -57
11.
19.275 -74.63
12.
4,386 -5,119
13.
2.35 -17.986
14.
2,984 - 151
15.
Cost:$32.50 Discount:$1.79 Final Price:
16. Temp:67° F Change:18° warmer Final:
17. Altitude: 7,380 ft Fall: 3,200 ft Final:
18.
Cost:$32.50 Tax:$2.08 Final Price:
19. Temp: 17° C Change: 28° colder Final:
20. Altitude:300 m Rise:7,250 m Final:
Change into a decimal. 21.
2 5
22.
1 4
23.
3 8
24.
1 9
25.
7 8
26.
1 6
Assignment 1.3a
26
Divide. Example: See examples in section 1.3 27.
7 234
28.
5 135
29.
11 589
30.
.04 56.3
31.
.8 42
32.
2.1 151.2
34.
19 20
35.
15 45
Change into a percent. 33. 129
Using the chart, find out how much money was spent if the total budget was $1600. 36.
Insurance
Find the following: 39. Price: $30.00 Tax rate: 6% Tax:
37.
House
38.
Fun
40.
Attendees: 2,300 Percent men: 40% Men:
41.
Students: 4 Number of B’s: 3 Percent of B’s:
Preparation. 42. Go to providentliving.org and read the “One for the Money” and “All is Safely Gathered In” pamphlets. Be ready to share thoughts and notes with your group.
Assignment 1.3a
27
Answers: 38,648 1.
31. 32.
52.5
2.
144.213
3. 4.
5,834,960.5227
33. 34.
75%
9
5.
-9
35.
33.3%
6.
-3
36.
$144
-13
7. 8. 9.
72
95%
$752
-22
37. 38. 39.
10.
177
40.
920 men
11. 12.
-55.355
41. 42.
75%
13. 14.
-15.636
15. 16.
$30.71
17.
4180 ft
-47
-733
$160 $1.80
Discuss it together.
2833
85° F
$34.58
18. 19.
-11° C
20. 21.
7550 m
22. 23.
.25
24.
.1 .875
25. 26.
.4
.375
.16
27. 28.
27
29. 30.
53 11 or 53.54 or 53 R6 1407.5
33 73 or 33.428571 or 33 R3 6
Assignment 1.3a
28
Section 1.3 Exercises Part B Add. 1.
36,851 + 3,197
2.
153.29 + .922
3.
8,434,916.7 + 54.3527
Subtract. 4.
9 - (-3) =
5.
-18 – 32 =
6.
-14 –(-19) =
7.
-18 + 6 =
8.
15 – 47 =
9.
-24 – 8 =
10.
754 -57
11.
29.84 -64.643
12.
4,786 -5,919
13.
2.35 -13.946
14.
23,754 - 4,151
15.
Cost:$32.50 Discount:$5.79 Final Price:
16. Temp:67° F Change:28° warmer Final:
17. Altitude: 4,380 ft Fall: 2,230 ft Final:
18.
Cost:$33.50 Tax:$2.18 Final Price:
19. Temp: 27° C Change: 48° colder Final:
20. Altitude:300 m Rise:2,250 m Final:
Change into a decimal. 21.
4 5
22.
2 9
23.
24.
1
25.
5
26.
8
6
5 8
1 10
Assignment 1.3b
29
Divide.
27.
7 434
28.
6 135
29.
12 789
30.
.04 56.347
31.
.6 453
32.
3.1 125.12
Change into a percent. 33. 127
34.
17 20
35.
15 30
Using the chart, find out how much money was spent if the total budget was $1300. 36.
Food
Find the following: 39. Price: $77.20 Tax rate: 6% Tax:
37.
Car
38.
Fun
40.
Attendees: 2,400 Percent men: 79% Men:
41.
Students: 12 Number of B’s: 11 Percent of B’s:
Begin “Budget and Expenses” Portfolio Project 42. Make sure all members of the group have seen the pattern of budget and expense reports found in “All is Safely Gather In” and “One for the Money.” Begin a monthly budget and record of your expenses that will continue through the remainder of the semester. Commit to reporting to your group and receiving reports when all have created a spreadsheet titled, “Budget and Expenses.”
Assignment 1.3b
30
Answers: 40,048 1.
31. 32.
755
2.
154.212
3. 4.
8,434,971.0527
33. 34.
58.3%
12
5.
-50
35.
50%
6.
5
36.
$260
-12
7. 8. 9.
40.361…
85%
$182
-32
37. 38. 39.
10.
697
40.
1896 men
11. 12.
-34.803
41. 42.
91.67%
13.
-11.596
14.
19,603
15. 16.
$26.71
17.
2150 ft
18. 19. 20.
$35.68 -21° C
21.
.8
22. 23. 24. 25.
.2
26.
.1
27. 28.
62
29. 30.
65.75 1408.675
-32
-1,133
$130 $4.63
Submit it to your teacher later in this lesson.
95° F
2550m
.625 .125 .83
22.5
Assignment 1.3b
31
Section 1.3 Exercises Part C Begin “Budget and Expenses” Portfolio Project. 1. Continue to record all expenses and income for the remainder of the course in a spreadsheet document. Round the following. 2. Round 54,454 to the nearest thousand. 3. Round 385,764,524.83 to the nearest million. Estimate the following. 4.
Add. 7.
71 × 3250.07
5.
538.9 × 2,892.07
6.
46,821 + 3,137
8.
756.29 + .522
9.
82 × .00000789
8,434.7 +54.3527
Subtract. 10.
115 - (-3) =
11. -19 – 320 =
12. -18 –(-151) =
13.
7.54 -57
14.
15.
16.
Cost:$44.50 Tax:$3.18 Final Price:
17. Temp: 48° C Change: 29° colder Final:
298.4 -64.643
3,784 -5,919
18. Altitude:300 m Fall:2,250 m Final:
Change into a decimal. 19.
1 20
20.
4 9
21.
2 3
Assignment 1.3c
32
Divide.
22.
8 434
23.
6 185
24.
14 689
25.
.02 56.347
26.
.6 553
27.
.31 175.12
Change into a percent. 28. 37 40
29.
38 50
30.
27 25
Using the chart, find out how much money was spent if the total budget was $1354. 31.
Insurance
Find the following: 34. Price: $75.37 Tax rate: 6% Tax:
32.
Car
33.
Fun
35.
Attendees: 2,413 Percent men: 39% Men:
36.
Students: 15 Number of B’s: 11 Percent of B’s:
Assignment 1.3c
33
Answers: Titled “Budget and Expenses” and save 1. document on your computer. You will turn it in to your teacher in this lesson. 54,000 2.
3.
386,000,000
4. 5.
About 210,000 About 1,500,000
31.
$108.32
32. 33.
$203.10
34. 35.
$4.52
About .00064
6. 7. 8.
49,958
$135.40
941 men 73.3%
36.
756.812
9. 10.
8489.0527
11. 12. 13.
-339
14. 15.
233.757
16.
$47.68
17. 18.
19° C -1950m
19. 20.
.05
21. 22.
.6
23. 24. 25.
30.83
26. 27.
921.6
28. 29.
92.5% 76%
30.
108%
118
133 -49.46
-2135
.4
54.25
49.214… 2817.35
564.903…
Assignment 1.3c
34
Section 1.4 Fractions
DEFINITIONS & BASICS 1) Numerator – the top of a fraction 2) Denominator – the bottom of the fraction 3) Simplify – Fractions are simplified when the numerator and denominator have no factors in common. 4) One – any number over itself = 1. 5) Common Denominators – Addition and subtraction require like things. In the case of fractions, “like things” means common denominators. 6) Prime Factorization – Breaking a number into smaller and smaller factors until it cannot be broken down further.
LAWS & PROCESSES Prime Factorization – One of the ways to get the Least Common Denominator for adding and subtraction fractions that have large denominators is to crack them open and see what they are made of. Scientists get to use a scalpel or microscope. Math guys use prime factorization.
Addition of Fractions 1. Observation 2. Multiply the denominators 3. Prime factorization
1. Common Denominator 2. Add numerators 3. Carry by denominator
EXAMPLE Add
+
+
Step 1. The least common multiple of 4 and 2 is a 4, so we
replace the with an equivalent fraction, which is .
3+2 4 4
5
Step 2. Now that the denominators are the same, add the numerators. Section 1.4
35
Step 3. Carry the denominator across.
Changing from mixed numbers to improper fractions:
Changing them back again:
Subtraction of Fractions 1. 2. 3. 4.
Biggest on top Common Denominator; Subtract numerators Borrow by denominator Strongest wins
1. Observation 2. Multiply the denominators 3. Prime factorization
EXAMPLE
Do this:
-3
3
is bigger, so put it on top.
- 3
The common denominator is 9,
so change the
to a
.
-2
-2
Subtract the numerators. Borrow by denominator as needed.
Section 1.4
36
Multiplication of Fractions
Multiplication of Fractions 1. No common denominators 2. Multiply Numerators 3. Multiply Denominators
EXAMPLES
5 6
1 3
For multiplication don’t worry about getting common denominators
Multiply the numerators straight across
Multiply the denominators straight across
5 6
1 3
5 18
Section 1.4
37
Division of Fractions Division of Fractions 1. Improper Fractions 2. Keep it, change it, flip it. 3. Multiply.
EXAMPLES Divide 3
+
Turn the fractions into improper fractions
25 7
3 2
Keep the first fraction the same Change the division sign to a multiplication sign Flip the second fraction’s numerator and denominator Multiply straight across the numerator and denominator 4 2 3 7 3
Divide 1 2 1 4 3 + 5 4 4
2 5
7 4
2 5
4 7
1
75 14
Turn the fractions into improper fractions
KEEP the first fraction the same Change the division sign to a multiplication sign Flip the second fraction’s numerator and denominator Multiply straight across the numerator and denominator
Section 1.4
38
Now that you have had a little time to multiply fractions together and simplify them, you may have noticed one of the slickest tricks that we can do with fractions, and that is that we can actually do the simplification before we multiply them. Take for example:
Now, we can do this the normal way or we can try to notice if there is anything that we will be simplifying out later . . . and do that simplification before we multiply: Normal method:
New and improved slick method:
and now we try to simplify
and we try to see if any factors will cancel ahead of time
which probably took quite a while to get. So,
2×5
3×7
3×3×7
5×11
What I was hoping to show is that the same answer was obtained and the same cancelling was done, but if you are able to see it before you multiply, then you will be able to simplify in a much simpler way. Here is another example: the 4 and the 8 can simplify before we multiply:
This may seem like just a convenient way to make the problem go a bit quicker, but it does much more than that. It opens the door to a much larger world. Here is an example. If we travelled 180 miles on 12 gallons of gas, then we calculate the mileage by
= 15 miles per gallon.
Carrying that example just a bit further, what if gas were $3.2 per gallon? We can actually find how many miles we can drive for one dollar:
= 4.7 miles per dollar.
Another example:
Carpet is on sale for 15 dollars per square yard. How much is that in dollars per square foot (9 ft2 per yd2)? knowing wethe will be able to cancel the top anything thatdollars is the same onNow, the bottom wethat write multiplication so theanything yd2 willon cancel out,with leaving us with per ft2:
Section 1.4
39
1.666 3 5.00
Then cancel the yd2:
= dollars per square foot
-3 20 -18 20 -18
= $1.67 per square foot.
One more example:
A rope costs $15 for 8 feet. How much does is cost per inch? We want to get rid of feet and get inches, so we write the multiplication: = =
$.156 or 15.6 cents per inch.
.1562 32 5.00 -3 2 180 -160 200 -192 80
Here are a few numbers that will help you with the conversions: 12 in = 1 foot 1 yd = 3 ft 16 oz = 1 pound 60 minutes = 1 hour 2
60 seconds==11kilowatt minute 1000watts
2
1 yd = 9 ft
And also some exchange rates with the American dollar as they were sometime in 2010: 1 Mexican Peso = $0.08 1 Euro = $1.30 1 British Pound = $1.50 1 Brazilian Real = $0.55
Section 1.4
40
Section 1.4 Exercises Part A Find 4 different names for each fraction: Example: 3
3 6 9 12 15 30 3,000 , , , , , , ... 11 22 33 44 55 110 11,000
11
1.
2.
3 7
3.
2 3
4.
7 11
4 9
Simplify each fraction. 5.
36 52
6.
27 36
7.
16 56
8.
10 12
9.
15 45
10.
120 280
13.
10 12
Create each fraction with a denominator of 36. 5 11. 16 12. 9
Add or Subtract. Simplify. Common denominator
Example: 5 83 − 13 14
Example: 1 3 + = 2 7 7 6 13 + = 14 14 14
- 13 14
- 13 82
5 83
5 83
Borrow from the 13.
- 12 108
5 83 - 7 78
Swap to subtract. Answer is negative
14.
2 2 + = 5 3
17.
1 3
+
7 12
=
15. 18.
1 5 + = 4 8
13 3 + 4 5 4
6
=
16.
7 3 − = 30 25
19.
9
7 10
−
31
=
5
Assignment 1.4a
41
20.
3 149
−
6 76
Fill out the table. Mixed 8 23. −7 9 24.
21.
=
4 72 + 9 23
22.
=
12 85 − 9 34
=
Improper
3 15
25.
43 8
26.
51 4
Find the multiplicative inverse or reciprocal of each number. Example: 5 8
8 5
27.
4 7
28.
2 9
29.
- 107
30.
7 8
31.
- 56
32.
13
33.
13 42
34.
7 3
Divide. Example: 2 83 ÷ 54 =
2 83 × 54
Multiply by reciprocal
=
Change to improper fraction
19 5 × = 8 4 19 5 95 × = 8 4 32
31 or 2 32
Multiply straight across.
35.
2 1 ÷ = 5 3
36.
1 3 ÷ = 4 8
38.
3 7 ÷ = 8 12
39.
2 34 ÷ 7 16
41.
7 45 ÷ 109
42.
7 ÷ 8
=
9 23
=
=
37.
5 3 ÷ = 6 8
40.
5 75 ÷ 3 23
43.
2 16
÷
=
3 = 8
Preparation. 44. If you drive 280 miles on 12 gallons of gas, how many miles per gallon do you get? 45. If you drive 280 miles on 12 gallons of gas, and gas is $3.20 per gallon, how many miles per dollar do you get? Assignment 1.4a
42
Answers: 6 9 12 21 others... 1. 14 , 21 , 28 , 49 ,
31. 32.
−
33. 34.
42 13
2.
4 6
3. 4.
14 22
, ,
8 18
16 28 others... , 12 27 , 36 , 63 ,
5.
9 13
35.
6 5
6.
3 4
36.
2 3
12 others... , 96 , 10 15 , 18 , 21 33
28 44
35 55
, , others...
6 5
or
−
1 13
or 3 133
3 7
or 1 15
20
or 2 2
9 9 14
9
1 3
37. 38. 39.
10.
3 7
40.
120 77
11. 12.
6 36
41. 42.
26 3
13. 14.
30 36
2
7. 8. 9.
7 5 6
20 36
1 151 or 16 15
15. 16.
17 150
17.
11 12
18. 19.
18112 62
20. 21.
−
7 8
1 15
33 86 43 or 1 77
or 8 23
21 232
43. 44.
Discuss it together.
45.
Discuss it together.
52 9
or 5 79
7
3 143
13 20 21
22. 23.
−
24.
16 5
25. 26.
5 83
27. 28.
7 4
or 1 34
9 2
or 4 12
−
10
29. 30.
2 78 71 9
12
8 7
7
−
3
or 1 7
Assignment 1.4a
43
Section 1.4 Exercises Part B Create each fraction with a denominator of 24. 1.
2 3
2.
7 12
3.
40 48
5.
3 7 + = 4 9
6.
5 7 − = 12 8
8.
7 75 + 6 56
9.
2 58 − 9 35
Add or Subtract. Simplify. 4.
2 2 + = 5 7
7.
3 23 − 16 79
Fill out the table. Mixed 5 10. −2 9 11.
=
=
=
Improper
6 74
12.
35 8
13.
57 11
Find the multiplicative inverse or reciprocal of each number.
3 5
14.
15.
3 94
16.
- 125
7
17.
Divide. 18.
5 2 ÷ = 7 3
21.
2 23 ÷ 103
=
19.
3 6 ÷ = 4 7
22.
5 ÷ 8
4 12
20.
=
23.
1 4 ÷ = 6 9
- 2 73
÷
5 = 7
Change into a decimal.
24.
2 5
25.
1 4
26.
3 8
27.
1 9
28.
7 8
29.
1 6
Assignment 1.4b
44
Change into a fraction and simplify. Example .12
.12 = 12 (100th) =
12 100
= 253
simplify
30.
.5
31.
.7
32.
.45
33.
.52
34.
.75
35.
.6
Convert the following units. Example: Dog food cost $7.00 for 20 pounds. How many ounces per dollar? Solution:
45.71 ounces per dollar
36. Cereal cost $4.50 for 2 pounds. How much did it cost per ounce? 37. Fishing line costs $.02 per foot. How much would 200 yards cost? 38. I was able to drive 250 miles on 15 gallons of gas. If gas costs $3.10 per gallon, how many miles can I drive per dollar?
39. If my sprinkler sends out 5 gallons per minute, and if water costs $0.65 per 1000 gallons, how much does watering my lawn cost per hour?
40. How many Pesos are equal to 5 Euros? (1 Mexican Peso = $0.08, 1 Euro = $1.30) 41. How many Reals are equal to 7 Pounds? (1 Brazilian Real = $0.55, 1 British Pound = $1.50) 42. Create a visual chart for all arithmetic of decimals. Use plenty of examples. 43. Create a visual chart for all arithmetic of fractions including Unit Conversions.
Assignment 1.4b
45
Answers:
1. 2.
16 24
3. 4.
20 24
14 24
24 35
5.
or 1 19 36
6.
11 − 24
31. 32.
7 10
33. 34.
13 25
35.
3 5
36.
3 4
$0.14 per ounce $12.00
1
−
9 20
7. 8.
9 13 23 14 42
37. 38.
5.38 miles per dollar
9.
−
39 40
39.
$0.20 per hour
10.
23 − 9
40.
81.25 Pesos
11.
46 7
41.
19.09 Reals
12. 13.
4 83
42. 43.
Part of Portfolio
5
6
2 11
14. 15.
5 3
16.
−
17.
1 7
18. 19. 20. 21.
35 7 8
Part of Portfolio
9 31 12 5
6
22. 23.
3 8
8 89 5 36
- 3 25
24. 25.
.4
26. 27.
.375
28.
.875
.25
.1
.16
29. 30.
1 2
Assignment 1.4b
46
Section 1.4 Exercises Part C Exam 1 Review Exercises Estimate the product (round to the greatest value, then multiply). 1. 2,589,000×59.34 2. .005608×.07816 3. 3.847×2,564 Add. 4.
36,841 + 249.7
5.
723.3 + 39.7
6.
13 = 16 149 + 5 14
8.
-8 – (-11) =
9.
13 74 − 1 67
12.
4 11 × = 5 12
Subtract. Temp: -35.5° F Change: 13.4° warmer Final: Multiply. 10. Cost: $35.20 Quantity: 17 Total: Add or Subtract. Simplify. 13. 32 + 59 =
7.
16.
15 16
−
6 79
Fill out the table. Mixed 5 19. −3 7
11.
=
369×(-23) =
14.
5 11 + = 12 14
17.
5 109 + 13 18
22.
8 ÷ 9
25.
7 9
=
=
15.
5 5 − = 18 6
18.
12 94 − 9 142
23.
7 34
26.
2 7
=
Improper
20.
59 6
Divide. 21.
8 9
4 23
=
( 45 )
÷ −
=
5 18
Change into a decimal. 24.
5 12
Change into a fraction and simplify. 27.
.3
28.
.055
29.
.375
Assignment 1.4c
47
Divide. 30. 33.
7 485
.5 47.31
31. 34.
3 781
32.
43 673
.0004 562.4
35. A dishwasher uses about 1400 watts of power. If the power company charges 9 cents per kilowatt-hour, how much does it cost to run a dishwasher for 16 hours in the month? 36. I bought 8 yards of rope for $9.84. How much did it cost per foot? Change into a percent. 37. 24 25
38.
36 40
39.
17 50
Using the chart, find out how much money was spent if the total budget was $3200. 40.
Car
Find the following: 43. Price: $45.20 Tax rate: 7% Tax: Final Price:
41
House
42
Food
44.
Attendees: 239 Percent men: 29% Men:
45.
Price: $15.30 Discount: 30% Amount of discount: Final Price:
46. Round to the nearest ten: 583.872
Assignment 1.4c
48
Answers: About 180,000,000 1.
31. 32.
260 13 or 260.3
2.
About .00048
3. 4.
About 12,000
33. 34.
94.62
37,090.7
5.
763
35.
$2.02
36.
$0.41 per foot
6.
4 7
22
-22.1° F
7. 8. 9.
5 7
11
1,406,000
96%
37. 38. 39.
3
15 28 43 or 15.65116…
90% 34%
10.
$598.40
40.
$448
11. 12.
-8,487
41. 42.
$1504
43. 44.
$3.16, $48.36
45. 46.
$4.59, $10.71
13. 14. 15. 16.
11 15
2
1 18
1
23 84
5 − 9
8
7 18
17.
19
18. 19.
3 6326
20. 21.
9 56
22. 23.
4 21 11 - 9 16 or - 155 16
24.
.416
25. 26.
.7
27. 28.
3 10
29. 30.
8
$640
69 men
580
1 40
19 −
7
3 15 or
16 5
.285714
11 200 3
69 72 or 69.285714
Assignment 1.4c
49
Chapter 2: CALCULATORS and FORMULAS
Overview 2.1 Exponents and Calculator Usage 2.2 Variables and Formulas 2.3 More Variables and Formulas - Excel
50
Section 2.1 Exponents While we are on multiplication, did you know that there is some short hand? Remember when we started multiplication we did: 6+6+6+6+6+6+6+6+6 = 54 but we did it a bit shorter 9
9×6 = 54
There is a way to write multiplication in shorthand if you do the same thing over and over again: 2×2×2×2×2×2×2 = 128 7 For the shorthand we write 27 = 128. That little 7 means the number of times that we multiply 2 by itself and is called and exponent; sometimes we call it a power. Here are a couple more examples: 4 53= 125 = 49 72 2 = 16
Pretty slick. You won’t have to memorize them . . . yet, but you should be familiar enough with them to be able to recognize them. Some of the easiest to calculate are the powers of 10. Try these: 104= 10,000
8 10 = 100,000,000
10
3
= 1,000
EXAMPLE Evaluate 74
49 × 7 × 7 343 × 7 2401 Answer: 2401
Set up the bases, and then multiply each couple in turn.
Section 2.1
51
Order of Operations The last small note to finalize all your abilities in arithmetic is to make sure you know what you need to do when you have multiple operations going on at the same time. For example, 2+3×4–5 If you were to read that from left to right you would first add the 2 and the 3 to get 5 and then multiply by 4 to get 20 and then subtract 5 to get 15. Unfortunately, that doesn’t jive with what we have learned about what multiplication is. Remember that multiplication is a shorthand way of writing repeated addition. Technically we have: 2+3×4–5= 2 + 4 + 4 + 4 – 5 = 9. Ahh, now there is the right answer. It looks like we need to take care of the multiplication as a group, before we can involve it in other computations.Multiplication is done before addition and subtraction. Here is another one: 2 4×3 –7×2+4 Now remember that exponents are shorthand for a bunch of multiplication that is hidden, so we need to take care of that even before we do multiplication: 2 4 × 3 – 7 × 2 + 4 = Take care of exponents 4 × 9 – 7 × 2 + 4 = Take care of multiplication Add/Sub left to right. 36 – 14 + 4 = 22 + 4 = 26. Now division can always be written as multiplication of the reciprocal, so make sure you do division before addition and subtraction as well. Look at that. We have established an order which the operations always follow, and we need to know it if we are to get the answers that the problem is looking for: 1st – Exponents 2nd – Multiplication and Division (glues numbers together) rd 3 – Addition and Subtraction (left to right) Parentheses can change everything. We put parentheses when we intend on grouping (or gluing) numbers together manually. Though they all have the same numbers and operations, see the difference between these: 2 − 3 × 62
÷
2=
2 − (3 × 6 )
2
÷
2=
(2 − 3) × 6 2 ÷ 2 =
2 − 3 × 36 ÷ 2 =
2 − 18
2=
−
1 × 36 ÷ 2 =
2 − 54 = −52
2 − 324 ÷ 2 =
−
36 ÷ 2 = −18
2
÷
(2 − 3 × 6 )2 ÷ 2 = (2 − 18)2 ÷ 2 = (− 16 )2 ÷ 2 =
2 − 162 = −160
÷
256
=
2
128
Section 2.1
52
Section 2.1 Exercises Part A Calculator Usage Assignment On this assignment, you should use your calculator. Become familiar with it. It is now your friend! Estimate the product (round to the greatest value; then multiply). 1. 75,800×49.34 2. .004208×.06916 3. 4.447×7,164 Add. 4.
5.
37,291 + 348.23
Subtract. Temp: 85.3° F 7. Change: 130.4° colder Final: Multiply. 10. Cost: $38.40 Quantity: 27 Total: Find. 13. 3 =
8.
14 18 − 7 94
Fill out the table. Mixed 22. 7 114
6.
-5 –3 =
11.
14.
Add or Subtract. Simplify. 16. 34 + 49 = 19.
5.871 + 39.7
=
17
9
5
+5
23
=
9.
23 114 − 15 118
=
12.
- 25 × 16 11
23
441×29 =
15. 11 =
27 =
17.
5 8
20.
5 109
25.
5 6
28.
3 5
+
=
7 10
=
+ 19 18 =
18.
8 15
21.
4 58
3 − 5 16 =
26.
7 58
÷ 83 =
29.
2 9
−
7 9
=
Improper
23.
−
5 2
Divide. 24.
11 12 7 18
÷ 4 12 =
Change into a decimal. 27.
7 11
Assignment 2.1a
53
Change into a fraction and simplify. 30. .07 31.
.44
32.
.625
6 79
35.
57 6273
Divide. 33. 36. Evaluate 38.
34.
7 343
.5 4.731
−
37.
.004 967.4
39.
−
40.
−
41. Change 60 miles per hour into feet per second. (5280 feet = 1 mile) Change into a percent. 28 42. 30
43.
41 57
44.
37 100
Using the chart, find out how much money was spent if the total budget was $2437. 45.
Fun
Find the following: 48. Price: $380.50 Tax rate: 7% Tax: Final Price:
46.
Insurance
47.
Food
49.
Attendees: 48 Percent kids: 25% Kids:
50.
Students: 30 Number of A’s: 24 Percent of A’s: Assignment 2.1a
54
Answers: About 4,000,000 1.
31.
11 25
2.
About .00028
32.
5 8
3. 4.
About 28,000
33. 34.
49
5. 6. 7. 8.
45.571
35. 36. 37. 38.
110.0526…
37,639.23
22 14 23 -45.1° F -8
13 16 or 13.16 9.462 241,850 0
9.
7
39.
20
10.
$1,036.80
40.
-7
11.
12,789
41.
88 feet per second
42. 43.
93.3%
44. 45.
37%
1
7 36
46.
$219.33
1
13 40
47. 48.
$414.29
49. 50.
12 kids
7 11
12. 13.
-
14. 15.
729
16. 17. 18. 19. 20.
32 55
2187
19,487,171
−
11 45
6
49 72
25 401
21. 22. 23.
−
24. 25.
2 145 or
26. 27.
20 13 or .63
28.
.6
29. 30.
.2
71.9%
$316.81
$26.64, $407.14
80%
9 16
81 11
− 2 12 33 14
5 27 61 3
7 100
Assignment 2.1a
55
Section 2.1 Exercises Part B Add. 1.
57,831 + 348.23
Subtract. Temp: -85.3° F 4. Change: 130.4° colder Final: Multiply. Cost: $38.40 7. Quantity: 527 Total: Find. 9. 3 =
2.
4.83 + 39.7
3.
14 119 + 8 115 =
5.
-5 –53 =
6.
23 214 − 15 218 =
8.
12. (2.38) =
15 - 25 × 14
=
10.
37 =
11. (5.8) =
13.
(1.07) =
14. (1.12)1 =
15. If I place 2 cents on the first square of a chess board, 4 cents on the second square, and keep doubling the amount on each square, how much money will be on the 30 th square? Fill out the table. Mixed 16. 5 23
Improper −
17.
57 2
18. A product costs $7 for 20 pounds. How much is that in cents per ounce? 19. Change 17 Euros into pesos. (1 Mexican Peso = $0.08, 1 Euro = $1.30) 20. Change 60 miles per hour into feet per second. (5280 feet = 1 mile) Change into a percent. 21. 24 35
22.
72 64
23.
14 2000
Using the percentages, find out how much money was spent if the total budget was $2437. 24.
Fun – 12.3%
Find the following: 27. Price: $480.50 Tax rate: 7% Tax:
25.
Insurance – 7.9%
26.
Food – 38%
28.
Attendees: 388 Percent kids: 25% Kids:
29.
Students: 250 Number of A’s: 147 Percent of A’s: Assignment 2.1b
56
Final Price:
30. For a savings account that begins with $100 and has a 5% interest rate, fill out the following table: Time Beginning Balance Interest earned Ending Balance year 1st 100 .05 × 100 = 5 105 year 2n 105 .05 × 105 = 5.25 110.25 3 110.25 .05 × 110.25 =5.51 115.76 4 115.76 5 6 7 8 9 10 11 12
31. For a savings account that begins with $100 and has a 6% interest rate, fill out the following table: Time Beginning Balance Ending Balance year 1st 100 100 × 1.06 = 106 year 2n 106 106 × 1.06 = 112.36 3 112.36 112.36 × 1.06 = 119.10 4 5 6 7 8 9 10 11 12
119.10
32. Discuss in your group why multiplying by .05 and then adding to the balance is the same as multiplying the balance by 1.05. 33. If a savings account started at $100 and earned 7% per year, how much would be in the account at the end of 12 years? 34. If a savings account started at $100 and earned 7% per year, how much would be in the account at the end of 22 years? 35. How can exponents be used to find the balance after many years? Assignment 2.1b
57
1.
58,179.23
31.
2.
44.53
32.
3. 4.
-215.7° F
33. 34.
-58
35.
5. 6.
12 year end balance - $201.22 ($201.23 also acceptable) 1 adds in the beginning balance and .05 adds in the 5% $225.22 $443.04 #34 can be done by 100 × (1.07)
$20,236.80
7. 8.
9.
243
10.
1,369
11.
195.112
12. 13.
5.6644
14. 15. 16.
3.896
17.
-
18. 19. 20.
2.19 cents per ounce 276.25 Pesos
21. 22.
68.6%
23.
0.7%
24. 25. 26. 27.
$299.75
28.
97 kids
29. 30.
58.8% 12 year end balance - $179.59 ($179.60 also acceptable)
6.214
$10,737,418.24
88 feet per second
112.5%
$192.52 $926.06 $33.64; $514.14
Assignment 2.1b
58
Section 2.1 Exercises Part C 1. Find three different places to save your money. Report the interest rates to your group, and receive their reports. Find. 2.
4=
3.
87 =
4.
(2.7) =
5.
(5.38)
6.
(1.06)
7.
(1.11)
Fill out the table. 8.
Mixed 5 25
Improper
9.
−
37 3
10. If I place 1 cent on the first square of a chess board, 2 cents on the second square, and keep doubling the amount on each square, how much money will be on the 20 th square?
11. A product sells for $2.50 per square foot. How much is that per square yard? 12. Change 400 Pesos into Pounds. (1 Mexican Peso = $0.08, 1 British Pound = $1.50) 13. Change 50 miles per hour into feet per second. (5280 feet = 1 mile) Change into a percent. 14.
15.
16.
Using the percentages, find out how much money was spent if the total budget was $287. 17.
Fun – 17.3%
Find the following: 20. Price: $80.40 Tax rate: 7% Tax: Final Price:
18.
Insurance – 6%
21.
Attendees: 388 Percent kids: 35% Kids:
19.
Food – 84%
Assignment 2.1c
59
22. For a savings account that begins with $350 and has a 5% interest rate, fill out the following table and place the entries in the “Life Plan” spreadsheet on Sheet 2: Time year 1st n 2year 3 4 5 6
Beginning Balance Ending Balance 350 350 × 1.05 = 367.50 367.50
7 8 9 10 11 12
23. If a savings account started at $300 and earned 7% per year, how much would be in the account at the end of 22 years?
24. For a savings account that begins with $100 and has a 6% interest rate and to which you are able to add $25 per year, fill out the following table and place it on Sheet 2 of your Life Plan spreadsheet: Time Beginning Balance Ending Balance year 1st 100 100 × 1.06 + 25 = 131 n 2year 131 131 × 1.06 + 25 = 163.86 3 163.86 163.86 × 1.06 + 25 = 4 5 6 7 8 9 10 11 12
25. If a savings account started at $200 and earned 7% per year, how much would be in the account at the end of 12 years if you are able to add $40 per year?
Assignment 2.1c
60
2.
Complete when all reports are done. 1024
3.
7569
4. 5.
143.489
6. 7. 8.
4.29
9.
−
10.
$5,242.88
11.
$22.50 per square yard
12.
21.33 pounds
13.
73.3 feet per second
14.
66.7%
15. 16.
54.4%
17.
$49.65
18. 19. 20.
$17.22 $241.08
21. 22.
(135.8) 136 kids
23. 24. 25.
$1329.12
1.
28.944
3.88
.02%
$5.63; $86.03
$628.55
$622.97 $1,165.98
Assignment 2.1c
61
Section 2.2 Variables and Formulas
Variables and Formulas DEFINITIONS & BASICS
1) Variables: These symbols, being letters, actually represent numbers, but the numbers can change from time to time, or vary. Thus they are called variables. Example: Tell me how far you would be walking around this rectangle. 24 ft 15 ft 15ft
24 ft It appears that to get all the way around it, we simply add up the numbers on each side until we get all the way around. 24+15+24+15 = 78. So if you walked around a 24ft X 15ft rectangle, you would have completed a walk of 78 ft. I bet we could come up with the pattern for how we would do this all of the time. Well, first of all, we just pick general terms for the sides of the rectangle: length width
width
length Then we get something like this: Distance around the rectangle = length + width + length + width Let's try and use some abbreviations. First, “perimeter” means “around measure”. Substitute it in: Perimeter = length + width + length + width Let's go a bit more with just using the first letters of the words: P=l+w+l+w Notice now how each letter stands for a number that we could use. The number can change from time to time. This pattern that we have created to describe all cases is called a formula.
Section 2.2
62
2) Formula: These are patterns in the form of equations and variables, often with numbers, which solve for something we want to know, like the perimeter equation before, or like: Area of a rectangle:
A=B×H
Volume of a Sphere:
Pythagorean Theorem:
Through the same process we can come up with many formulas to use. Though it has all been made up before, there is much to gain from knowing where a formula comes from and how to make them up on your own. I will show you on a couple of them.
Distance, rate If you were traveling at 40mph for 2 hours, how far would you have traveled? Well, most of you would be able to say 80 mi. How did you come up with that? Multiplication: (40)(2) = 80 (rate of speed) ⋅ (time) = distance or in other words:
rt = d where
r is the rate t is the time d is the distance
Percentage If you bought something for $5.50 and there was an 8% sales tax, you would need to find 8% of $5.50 to find out how much tax you were being charged. .44 = .08(5.50) Amount of Tax = (interest rate) ⋅ (Purchase amount) or in other words:
T = rP Where T is tax r is rate of tax P is the purchase amount.
Interest This formula is a summary of what we did in the last section with interest. If you invested a principal amount of $500 at 9% interest for three years, the amount in your account at the end of three years would be given by the formula: A = 500(1.09)3 = $647.51
Section 2.2
63
A = P(1 + r)Y where A is the Amount in your account at the end P is the principal amount (starting amount) r is the interest rate Y is the number of years that it is invested.
Temperature Conversion Most of us know that there is a difference between Celsius and Fahrenheit degrees, but not everyone knows how to get from one to the other. The relationship is given by:
C = 5 (F – 32) 9
where F is the degrees in Fahrenheit C is the degree in Celsius
Money If you have a pile of quarters and dimes, each quarter is worth 25¢ (or $.25) and each dime is worth 10¢ ($.10), then the value of the pile of coins would be:
V = .25q + .10d where V is the Total Value of money q is the number of quarters d is the number of dimes
3) Common Geometric Formulas: Now that you understand the idea, these are some basic geometric formulas that you need to know:
P is the perimeter l
w Rectangle
P = 2l + 2w A = lw
l is the length
w is the width A is the Area
Section 2.2
64
P = 2a + 2b
b
P is the perimeter a is a side length
a h
b is the other side length
A = bh
Parallelogram
A is the Area h is the height
P is perimeter
b a
h
d
P = b+a+B+d
b is the shorter base B is the longer base
B
A = 1 h(B+b) 2
a is a leg d is a leg A is the Area
Trapezoid
h is the height
P = s1+s2+s3 A = 1 bh 2
h b Triangle
P is the perimeter s is a side
A is the Area b is the base h is the height a is one angle
b c
a + b + c = 180°
b is another angle c is another angle
a Triangle
Section 2.2
65
SA = 2lw+2wh+2lh H w l
SA is the Surface Area l is the length
w is the width
V = lwh
h is the height
Rectangular Solid
V is C is volume the Circumference or Perimeter
C = 2πr r
is a number, about 3.14159 . . . it has a button on your calculator π
A = πr2
r is the radius of the circle
LSA = 2πrh
A is the area inside the circle. LSA is Lateral Surface Area or area just on the sides
Circle
r
2 h
SA =2πrh+2πr
π
is a number, about
3.14159 . . . it has a button on your calculator
r is the radius of the circle Cylinder
V = πr2h
h is the height SA is total surface area
V is Volume
Section 2.2
66
h
l
r
LSA = πrl
LSA is Lateral Surface Area or the area just on the sides
SA = πr2+ πrl
π
V = 1 πr2h 3
Cone
is a number, about 3.14159 . . . it has a button on your calculator
r is the radius of the circle h is the height l is the slant height
SA is total surface area
r
SA = 4πr2
SA is the surface area is a number, about 3.14159 . . . it has a button on your calculator π
V = 4 π r3 3
r is the radius Sphere
V is the Volume
Section 2.2
67
Section 2.2 Exercises Part A Add or Subtract. Simplify. 1.
6 78 − 13 83
4.
3.7 9.574
=
2.
7 125 + 187 34
5.
6000 254.7
=
3.
21 56
6.
.008 37 .65
2 − 97 15 =
Divide.
7. If a wood floor costs $4.50 per square foot, how much is that per square yard? 8. How much does it cost to run a 700 watt microwave for 17 hours if the power company charges 12 cents per kilowatt-hour?
Find the following: Price: $39.48 9. Tax rate: 5% Tax: Total Price:
Evaluate the following: 12. 4 ⋅ 3 − 8(9)
15.
45(7.8)
10.
Price: $2,736.00 Percent off: 35% Amount saved: Final Price:
11.
Birds: 140 Black : 47 Percent of black birds:
13.
5⋅ d ⋅7 ⋅ p
14.
5(3 − 9) − 2 3 ⋅ (5 + 4)
16.
3⋅7 ⋅ m ⋅ 2
17
2(3 )+5(4)+8 ⋅ m
Find the perimeter of the following shapes: 18.
19.
20.
17
15
14 9
11
19
19
5 k-12 13
t+3
8
r
Evaluate the following when m = 3, n = 7, t = 15, and a = 4. 21.
3t - 7
22.
2(n+9)
23.
3 ⋅ 28a + m2
Assignment 2.2a
68
24.
12 – a
25.
m – 12
26.
2n – 3a + 5t
Use the formula for distance, rate and time to calculate the distance. Example: r=3 t = 14 d=
27.
r=7 t = 15 d=
Formula is found in section 2.3: rt = d 3(14) = d 42 = d
28.
r = 55 t = 7.2 d=
29.
r = 45 t = 2 13 d=
Use the formula for angles in a triangle to calculate the measure of the remaining angle. 30.
a = 73° b = 24° c=
31.
a = 38° b= c = 59°
32.
a= b= 24° c= 48°
Use the formulas for Money totals (you may have to make up your own) when q stands for quarters (1 quarter = $0.25), d for dimes (1 dime = $0.10), n for nickels (1 nickel = $0.05) and p for pennies (1 penny = $0.01). 33.
q=9 d = 12 V=
34.
p = 19 d = 17 V=
35.
n = 37 q = 23 V=
38.
F = -23° C=
Use the formulas for Temperature Conversion. 36.
F = 75° C=
37.
F = 15° C=
Preparation: 39. If the formula for area of a circle is
A=πr2
What is the area of a circle with radius 7?
40. Where did π come from? (Try finding out using dictionaries or the internet)
Assignment 2.2a
69
Answers: − 6 12 1.
31.
83°
2.
195
1 6
32.
108°
3.
− 75
3 10
33.
$3.45
4.
2.5876
34.
$1.89
5.
.04245
35.
$7.60
6.
4,706.25
36.
23.9° C
7.
$40.50 per square yard
37.
-9.4° C
8.
$1.43
38.
-30.6° C
9.
$1.97 and $41.45
39.
Discuss together.
10.
$957.60 and $1,778.40
40.
Discuss together.
11.
33.6%
12.
-60
13.
35dp
14.
-102
15.
351
16.
42m
17.
38 + 8m or 8m + 38
18.
58 + t or t + 58
19.
k + 21 or 21 + k
20.
42 + r or r + 42
21.
38
22.
32
23.
30
24.
-52
25.
-9
26.
77
27.
105
28.
396
29.
105
30.
83°
Assignment 2.2a
70
Section 2.2 Exercises Part B Evaluate the following when p = 8, r = -7, t = 1.
12 + a
2.
2 3
, and a = 3.
10a - 12 3r
3.
5r – 7p + 6t
Use the formula for Interest to calculate the amount in the account at the end of the time period. 4.
P = 520 r = 6.2% Y=4 A=
5.
P = 35,000 r = 6% Y = 9.3 A=
6.
P = 200 r = 8.9% Y=7 A=
Use the formulas for Money totals (you may have to make up your own) when q stands for quarters, d for dimes, n for nickels and p for pennies. 7.
q = 25 p=p p=p 8. 9. d = 17 d = q-13 q=q n = 15 V= n = q+7 V= V= Use the formula for Temperature Conversion to calculate the temperature in degrees Celsius.
10.
F = 300° C=
11.
F = -45° C=
12.
F = 102° C=
15.
r=3 l=8 LSA =
18.
Two angles are 37° and 81°; what is the third?
21.
b=7 B = 15 a = 12
Use the formulas for a cone to calculate the missing value. 13.
r=6 h = 11 V=
14.
r=9 l=5 SA =
Use the formulas for a triangle to calculate the missing value. 16.
b = 24 h=5 A=
17.
b = 15 h=4 A=
Use the formulas for a trapezoid to calculate the missing value. 19.
b=7 B = 10 h=7
20.
b=9 B = 15 h=3
Assignment 2.2b
71
A=
A=
d=8 P=
Use the formulas for a rectangular solid to calculate the missing information. 22.
l=6 w=9 h=7 SA =
23.
l=4 w = 15 h=7 SA =
24.
l=6 w = 14 h=2 V=
25.
8y + 5y
26.
4a – 9 + 4a
27.
16r – 5t + 3t + 12r
28.
7(x – 5) +15x
29.
7t + 4(t + 12)
30.
8 – 6(7 – 4t) +4t
31.
8 – 12x + 5 + 3x
32.
7x – 5x – 9x + 13x
33.
13xy + 7x(6y – 4)
Simplify.
As a group, discuss the following: 34. If the radius and height in #13 are in meters, what is the unit of the Volume? 35. If the bases and height in #19 are in inches, what is the unit of the Area? 36. If all the sides in #21 are measured in millimeters, what is the unit of the Perimeter? 37. If the radius and height in #15 are in miles, what is the unit of the Lateral Surface Area? 38. If all the sides in #24 are measured in yards, what is the unit of the Volume?
Assignment 2.2b
72
Answers: 39 1.
31.
-9x + 13
2.
−
6 7
32.
-2x + 8x
3.
-87
33.
55xy – 28x
4.
$661.46
34.
m – cubic meters
5.
$60,174.51
35.
in – square inches
6.
$363.27
36.
mm – millimeters
7.
$8.70
37.
mi – square miles
8.
V = .01p + .1(q-13)
38.
yd – cubic yards
9.
V = .01p + .3q + .35
10.
148.9°
11.
-42.8°
12.
38.9°
13.
132π or 414.69
14.
126π or 395.84
15.
24π or 75.4
16.
60
17.
30
18.
62°
19.
59.5
20.
36
21.
42
22.
318
23.
386
24.
168
25.
13y
26.
8a – 9
27.
28r – 2t
28.
22x – 35
29.
11t + 48
30.
28t – 34
Assignment 2.2b
73
Section 2.2 Exercises Part C Please label everything with the correct units. Evaluate the following when f = 5, r = -7, t = 1.
6t – f
2.
2 3
, and a = -2.
10a - 12 2f
+t
3.
2fr – 31a + 15a
6.
P = € 1,300
Use the formula for Interest. 4.
P = $15,000
5.
r = 6.2% Y=7 A=
P = £ 2,300 r = 6% Y = 8.7 A=
r = 8.9% Y=7 A=
Use the formulas for Money totals (you may have to make up your own) when q stands for quarters, d for dimes, n for nickels and p for pennies. 7.
q = t+5 p = 15 p = h+9 8. 9. d=m d=9 q=7 n = 13 V= n = x - 20 V= V= Use the formula for Temperature Conversion to calculate the temperature in degrees Celsius.
10.
F = -20° C=
11.
F = 59° C=
12.
F = 32° C=
15.
r = 3 yd h = 8 yd LSA =
18.
Two angles are 45° and 79°; what is the third?
21.
b = 12 ft B = 25 ft a = 13 ft d = 17 ft P=
Use the formulas for a cylinder to calculate the missing value. 13.
r = 6 in h = 12 in V=
14.
r=9m h=5m SA =
Use the formulas for a triangle to calculate the missing value. 16.
b = 6 ft h = 5 ft A=
17.
b = 15 cm h = 4 cm A=
Use the formulas for a trapezoid to calculate the missing value. 19.
b = 9 km B = 11 km h = 7 km A=
20.
b = 8 mm B = 15 mm h = 5 mm A=
Assignment 2.2c
74
Simplify. 22.
9y – 11y
23.
10a – 2b + 4a – 9b
24.
8(r – 7t) + 8(t +6r)
25.
2(x – 5) +7
26.
8m+ 4(m + 15t)
27.
9 – 5(6 – 9p) +4p
28.
8x – 34x + 9x + 10x
29.
12x4 – 5x – 4x4 + 13x
30.
3xy – 7x(5y – 4m)
31. If tile costs $1.50 per square foot, how much is that per square yard? 32. How much does it cost to run an 800 watt microwave for 17 hours if the power company charges 11 cents per kilowatt-hour? 33.
Change 3 Euros into Pesos. (1 Euro = $1.30, 1 Mexican Peso = $0.08)
34.
Change 66 feet per second into miles per hour. (5280 feet = 1 mile)
Assignment 2.2c
75
Answers: -121 1.
2.
−
3. 4.
-38
5.
£ 3,818.47
6.
€ 2,361.23
38 15
or
−2
8 15
$22,854.03
31. 32.
$13.50 per yd
33. 34.
48.75 Pesos
$1.50
45 miles per hour
.25t + .1m + 1.9
7. 8. 9.
$1.05 .01h + .05x + .84
10.
-28.9° C
11. 12.
15° C
13. 14.
1,357.17 in
15. 16.
150.8 yd
17.
30 cm
0° C
791.68 m
15 ft
56°
18. 19. 20. 21.
70 km 115 2
or 57.5 mm2
67 ft
22. 23.
-2y
24.
56r – 48t
25. 26.
2x – 3
27. 28.
49p – 21
14a – 11b
12m + 60t
-24x +17x 8x4 + 8x
29. 30.
-32xy + 28xm
Assignment 2.2c
76
Section 2.3 – More Formulas A calculator is a beautiful thing. You have been able to use one for a short time now and have probably enjoyed it considerably when compared to doing all of the math by hand. You are now ready to take another step with a much more powerful calculator – a computer. During this lesson, you are going to learn the basics of spreadsheets and how to make a computer do the calculations for you. During this discussion, we will use Microsoft Excel as the spreadsheet, but similar functions can be done in spreadsheets that are available at no cost such as OpenOffice – Calc.
Microsoft
Excel Basics
Microsoft Excel is spreadsheet software that allows you to perform calculations that help solve math problems in this course. You supply key figures and Excel automatically makes the calculations for you. Open Excel on your computer by clicking Start then Programs then Microsoft Excel. The main spreadsheet in Excel will appear. The spreadsheet is divided into cells each of which has a column and row address. Excel identifies columns by alphabetical letters and rows by numbers. The first cell in the upper left corner is A1. The cell to the right of it is B1 and so forth. The cell below A1 is A2 and so forth. You enter numbers, formulas, or words into the cells. Use the following guidelines as you enter data into Excel. It is easiest to enter numerical data in cells by using the number keypad on your keyboard. Be sure the Num Lock key is pressed and the Num Lock light is on. The number keypad also has four arithmetic functions you will need which are + (add), (subtract), * (multiply), and / (divide). It also has the numbers and an enter key so you can enter data rapidly using the keypad. Enter the = (equal) sign in the cell before you perform any calculation in Excel. This tells Excel you want it to perform a calculation. Use the following guidelines to format data in Excel. Never enter dollar signs ($) or commas (,) when entering data in Excel. Enter these by formatting the cell. Right click the cell or range of cells and select Format Cells. This opens a window that allows you to set the format in number, general, currency, percent, etc. You can set the number of decimal points you want to use and you can set alignment, font, etc. in this window. The cell format already has been set in most of the exhibits you will be using in this course. TIP: You can also format data in cells by clicking the cell or range of cells then clicking the appropriate symbol on the formatting tool bar. Section 2.3
77
Lifelong Income Example – Beginning Salary You can estimate your lifelong income using Excel To determine Lifelong Income do the following: 1. Enter the beginning hourly rate you will earn in your first job after you graduate in cell E3, for example $15.00. 2. Enter the number of hours you will work in a year in cell E5 as follows: =40*52 where 40 is the number of hours per week and 52 is the number of weeks in a year. 3. Press enter. Excel automatically multiplies 40 hours per week times 52 weeks per year and provides the result or 2080 working hours per year. 4. To calculate your first year salary in cell E7, enter (a) the equal sign, (b) click cell E3 (rate per hour) then enter * (multiplication sign) and (c) click cell E5 (hours per year). 5. Press enter. Excel calculates your first year’s income at $31,200. These entries are illustrated below:
Yearly Income Calculation – Format Rate Per Hour:
In Cell E3
Enter 15
Results $15.00
Hours Per Year
E5
=40*52
2080
Income - First Year of Employment (Beginning): E7
=E3*E5
$31,200.00
When you click on a cell that has a calculation set up, the formula for that cell appears in the formula line (to the right of the = sign) at the top of the page. For example, the formula line for the calculation performed in step 5 above would be: =E3*E5 Once your calculations are in place, Excel can save you time and effort if changes are required. If you were to change the beginning rate per hour to $10.00 and you have used the cell addresses in each of your formulas, Excel will recalculate all of the numbers and give you the new values. Try it. Enter “10” in E3 and watch what happens to the Income.
Section 2.3
78
To help get you used to formulas in Excel and how they work, we will use some of our familiar formulas from last week:
Circle Example Pick a cell where you will enter the radius – say B2. Put “2” in B2 as a starting radius. Then we write the formula for area in a cell next to it – C2. Remember the formula for area of a circle is
So, in C2 we write “=PI()*B2^2” π
variable for radius
exponent in Excel
Then you will notice that the area 12.56637 pops up in C2. Change the radius to “7” and you will be able to see that the area automatically changes. Nifty, isn’t it? You can change the radius to any number you would like and the area calculation will automatically update. Now, the power of Excel doesn’t stop just there. We can see the areas of a whole bunch of radii at the same time. List out several numbers in the cells beneath the “7” in B2. Now, if you copy the formula from C3 and paste it in C4, C5, C6, etc. you will notice that we can make a whole table of areas. If you label the columns, then others that see yourspreadsheet will be able to tell what you did. It should look something like this:
Temperature Conversion Example Make a column of numbers that are temperatures in Fahrenheit starting with cell C10. Then type in the formula that converts Fahrenheit to Celsius in D10: “=5/9*(C10 – 32)” Section 2.3
79
Copy and paste the formula into the cells next to the list of temperatures. See if it looks something like this:
Section 2.3
80
Section 2.3 Exercises Part A Using the formula for a rectangle and a calculator, fill out the following table: length width Perimeter Area 5 7 14 3 7.2 18.34 13 2.5 15 17 16 33 281 541.5 1.
2.
If the unit for length and width in #1 is inch, what are the units for Perimeter and Area?
3.
If the unit for length and width in #1 is centimeter, what are the units for Perimeter and Area?
Using the formula for a rectangle and a spreadsheet (Create a new file called Formula Practice), fill out the table in #1 using the formula abilities of the spreadsheet. 4.
5.
Using the formula for a circle and a calculator, fill out the following table: radius Circumference Area 3 12 5.1 17 4 38 114 6.
If the unit for radius in #5 is feet, what are the units for Circumference and Area?
7.
If the unit for radius in #5 is kilometer, what are the units for Circumference and Area?
Using the formula for a circle and a spreadsheet, fill out the table in #5 using the formula abilities of the spreadsheet. 8.
9.
Using the formula for a cone and a calculator, fill out the following table: radius height slant height LSA SA 3 4 5 5 12 13 15 8 17 24 6
7 8
Volume
25 10
Assignment 2.3a
81
10.
If the unit for radius, height and slant height in #9 is inch, what are the units for Lateral Surface Area, Surface Area, and Volume? 11.
If the unit for radius, height and slant height in #9 is centimeter, what are the units for Lateral Surface Area, Surface Area, and Volume?
12.
Using the formula for a cone and a spreadsheet, fill out the table in #9 using the formula abilities of the spreadsheet.
13.
Open your, “Budget and Expense” spreadsheet. Make sure that all budgets and expenses are updated. Using the “sum” formula, create cells that are the totals of your expenses and incomes. This spreadsheet will be submitted in your portfolio.
Assignment 2.3a
82
Answers: 1. length 5 14 7.2 13 15 16 281
width 7 3 18.34 2.5 17 33 541.5
2.
P – in; A – in
3.
P – cm; A – cm
4.
On Spreadsheet
Perimeter
Area
24 34 51.08 31 64 98 1645
35 42 132.048 32.5 255 528 152,161.5
5.
radius 3 12 5.1 17 4 38 114
Circumference
Area
18.85 75.40 32.04 106.81 25.13 238.76 716.28
28.27 452.39 81.71 907.92 50.27 4,536.46 40,828.14
6.
C – ft; A – ft
7.
C – km; A – km
8.
On Spreadsheet
9.
radius 3 5 15 24 6
height 4 12 8 7 8
slant height 5 13 17 25 10
10.
LSA – in ; SA – in ; V – in
11. 12.
LSA – cm ; SA – cm ; V – cm On Spreadsheet
13.
In Portfolio
LSA
SA 47.12 204.20 801.11 1884.96 188.50
Volume 75.40 282.74 1507.96 3694.51 301.59
37.70 314.16 1884.96 4222.30 301.59
Assignment 2.3a
83
Section 2.3 Exercises Part B Using the formula for a cylinder and a calculator, fill out the following table: radius height Surface Area Volume 5 7 14 3 7.2 18.34 13 2.5 15 17 16 33 281 541.5 1.
2.
If the unit for length and width in #1 is inch, what are the units for Surface Area and Volume?
If the unit for length and width in #1 is centimeter, what are the units for Surface Area and Volume? 3.
Using the formula for a cylinder and a spreadsheet, fill out the table in #1 using the formula abilities of the spreadsheet. 4.
Using the formula for a Sphere and a calculator, fill out the following table: radius Surface Area Volume 3 12 5.1 17 4 38 114 5.
6.
If the unit for radius in #5 is feet, what are the units for Surface Area and Volume?
7.
If the unit for radius in #5 is kilometer, what are the units for Surface Area and Volume?
8.
Using the formula for a sphere and a spreadsheet, fill out the table in #5 using the formula abilities of the spreadsheet.
84
9. Using a spreadsheet fill out the table for a savings account that has a beginning balance of $150 and grows at 7% with an additional $25 added at the end of each year: year Beginning Balance Ending Balance 1 150 150 × 1.07 + 25 = 185.5 2 185.5 185 × 1.07 + 25 =
. . .
use your calculator to make sure that the spreadsheet is calculating it correctly.
15
10. As a group, select a typical job that one of you anticipates having in the next five years. Then open a spreadsheet document and go through the lifelong income example in this section. How much money do you expect to earn over your lifetime?
Assignment 2.3b
85
Answers: 1. radius 5 14 7.2 13 15 16 281
height 7 3 18.34 2.5 17 33 541.5
Surface Area
2. 3.
SA – in ; V – in
4.
On Spreadsheet
5. radius 3 12 5.1 17 4 38 114
Volume
376.99 1,495.40 1,155.40 1,266.06 3,015.93 4,926.02 1,452,185.50
SA – cm ; V – cm
Surface Area
113.10 1,809.56 326.85 3,631.68 201.06 18,145.84 163,312.55
Volume
113.10 7,238.23 555.65 20,579.53 268.08 229,847.30 6,205,877.00
6.
SA – ft ; V – ft
7. 8.
SA – km ; V – km
9.
At the end of 15 years you should have $1,042.08 Complete when everyone can do it on their own.
10.
549.78 1,847.26 2,986.86 1,327.32 12,016.59 26,540.17 134,326,275.61
On Spreadsheet
Assignment 2.3b
86
Section 2.3 Exercises Part C Chapter 2 Exam Review Find the following: 1. 7 3 + 6 ⋅ 3 − 8(5)
2.
2⋅v⋅9⋅m
3.
6(3 − 7) − 4 2 ⋅ (7 + 4)
6.
18
Find the perimeter of the following shapes: 4.
5. 12
18 7
11
s-4
13
8 f+2 21
t+3
8
r-9
Find the following when p = -5, r = 7, t = 7.
12 – a
2 3
, and a = 4.
8.
7a - 12 4
9.
2r – 3p + 9t
12.
r = 36 feet per second
Use the formula for distance, rate and time. 10.
r = 6 m/h
11.
r = 65 km/h
1
t = 19 hours d=
t = 4.3 hours d=
t = 2 3 seconds d=
Use the formula for Interest. 13.
P = $2,800 r = 7% t=4 A=
14.
P = $5,000 r = 6% t=9 A=
15.
P = $300 r = 13% t=7 A=
Use the formulas for Money totals (you may have to make up your own) when q stands for quarters, d for dimes, n for nickels and p for pennies. 16.
q = 15 d = 27 V=
17.
p = 30 d = 25 V=
18.
p = 37 q = 23 n=7 V=
Assignment 2.3c
87
Use the formula for Temperature Conversion. 19.
F = 212 C=
20.
F = 98.6 C=
21.
F = -40 C=
23.
r = 9 ft l = 12.8 ft SA =
24.
r = 3 in l = 7.9 in LSA =
26.
l = 10.7 cm w = 4 cm A=
27.
l = 8.6 mm w = 9 mm P=
29.
r = 15 in A=
30.
r=7m C=
l = 4.2 mi w = 5 mi
33.
l = 6 km w = 8 km
Use the formulas for a cone. 22.
r=6m h= 7 m V=
Use the formulas for a rectangle. 25.
l = 3 yd w = 5 yd A=
Use the formulas for a circle. 28.
r = 4 in C=
Use the formulas for rectangular solid. 31.
l = 7 cm w = 2 cm
32.
h== 8 cm SA
h ==7mi V
h = km SA =
34. Create a Visual Chart on one side of a piece of paper for Chapter 2 material including information and examples relating to Calculator and Spreadsheet Usage and Formulas.
Assignment 2.3c
88
Answers: 321 1.
31.
172 cm
2.
18vm
32.
147 mi
3.
-200
33.
166 km
4.
30 + s + t
5.
f + 33
6.
r + 45
7.
-52
8.
4
9.
35
10.
114
11.
279.5
12.
84
13.
$3,670.23
14.
$8,447.39
15.
$705.78
16.
$6.45
17.
$2.80
18.
$6.47
19.
100° C
20.
37° C
21.
-40° C
22.
263.89 m
23.
616.38 ft
24.
74.46 in
25.
15 yd
26.
42.8 cm
27.
35.2 mm
28.
25.13 in
29.
706.86 in
30.
43.98 m
Assignment 2.3c
89
Chapter 3 ALGEBRA
Overview lgebra 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents
90
3+ “what” = 7? If you have come through arithmetic, the answer is fairly obvious: 4. However, if I were to ask something like: Linear Equations 2 times “what” plus 5 all divided by 7, then minus 6 = 5? There tends to be a little more difficulty in popping out the answer. The beauty of math is that it allows us to write down all of that stuff and then systematically make it simpler and simpler until we have only the number left. Wonderful. We start with the easy ones to find out all of the rules and then we will build up to the big ones.
Section 3.1
3+ “what” = 7 First, we need to adjust the fact that we are going to be writing “what” all the time. A very common thing is to put a letter in that place that could represent any number. We call that a variable. We replace the word “what” with “x” (or you could use p, q, r, f, m, l . . . ) So our equation becomes: 3+x=7 The whole goal of math is to find the number that makes that statement true. We already know that the number is 4. We would write: x=4 Now, look at what happened to our srcinal equation. Do you see that the right side is missing a 3 and the left side is now 3 lower as well. This gives us some insight into what we can do to equations! Try another one: x + 8 = 10 What number would make that statement true? If x were equal to 2, it would work. We write: x=2 Notice how we get the number that would work by subtracting that 8 from both sides of the equation. Let’s see if it works with some other equations: x–7=2 x – 3 = 10 With these two equations, the answers are: x=9 and x = 13 We got the answers by adding the 7 and the 3 to the right hand sides. This brings up a good point. In the first couple of equations that we did, we subtracted when the equation was adding. In the next two equations, we added when the equation was using subtraction. Let’s look at what happens when we start doing multiplication: 4x = 20. What number would work? That is right, 5. Section 3.1
91
x=5 What would you do to 20 to get 5? Divide by 4. Holy smokes! That is the exact opposite of what the equation is doing. Here is another: x =4 7 What number divided by 7 equals 4? That’s it, 28. We times 4 by 7 to get that answer. Multiplying by 7 is the exact opposite of dividing by 7. This leads us to a couple of conclusions that form the basis for everything we will do in Algebra:
1) When we want to get rid of numbers that are surrounding the variable, we need to do the opposite (technically called the inverse) of them.
A great way to think about these concepts is as though you have a balance that is centered on the equal sign. As long as you put the same thing on both sides, you remain balanced.
2) We can add, s ubtract, multiply, or divide both sides of an equation by any number and still have the equation work. Here is how it would work, one of each: x+7=11 4x=24 -7 -7 /4 /4 x=4
x=6
x-3=24 +3 +3 x = 27
x 5
=7 (5) (5) x = 35
You may ask why we go through all of that when the answers are obvious. The answer is that these problems will not be so easy later on, and we need to practice these easy ones so that when we get the hard ones, they crumble before our abilities. Now to some which are a little tougher. When we have one like this: 2x – 7 = 11 We could think about it long enough to find a number that works, and maybe you can do that, but I have to tell you that in just a little while we are going to have a problem that you won’t be able to do that with too quickly. So, let’s use what we learned to get rid of the 2 and the 7 so that x will be left by itself. If you remember the order of operations, you will remember that the 2 and the x are stuck together by multiplication, so we can’t get rid of the 2 until the 7 has been taken care of like this: 2x – 7 = 11 2x = 18 (we added 7 to both sides) x=9 (divided both sides by 2)
Section 3.1
92
To illustrate the idea of un-doing operations, I would like to try to stump you with math tricks. We begin. I am thinking of a number, and it is your job to guess what the number is. I am thinking of a number. I times the number by two. I get 10. Not too hard to figure out, you say? You're right. The answer is 5 and you obtained that by taking the result and going backwards. Try the next one:
I am thinking of a number. I times the number by 3. Then I subtract 5. Then I divide that number by 2. Then I add 4 to that. I get 18. What was the number I started with?
Aha. A little tougher don't you think? Well, If you think about it just one step at a time, then the thing falls apart. What number would I add 4 to to get 18? 14 (notice that it is just 18 subtract 4). We can just follow up the line doing the exact opposite of what I did to my number. Here you go:
Start with 18 Subtract 4 = 14 Multiply by 2 = 28 Add 5 = 33 Divide by 3 = 11.
That's it! Most of Algebra is summed up in the concept of un-doing what was done. I am thinking of a number. I times it by 4. Then I add 5. Then I divide by 9. Then I subtract 7. I get -2. What did I start with? This one is done the same way as the other one but I wanted to show you how you make that into an equation that will be useful in the rest of your math career. Instead of writing each
Section 3.1
93
step out, we construct an equation. We write it again but this time we will write the equation along with it: I am thinking of a number. I times it by 4. Then I add 5. Then I divide by 9.
We call that x. 4x 4x+5 4 x+5 9
Then I subtract 7.
4 x +5 − 9
7
I get -2. What did I start with?
4 x +5 − 9
7 = -2
That looks like a nasty equation, but it is done in exactly the same way. We just go backwards and un-do all of the things that were done to the srcinal number. We are using the rule that we can add, subtract, multiply or divide both sides of the equation by the same thing. I know you can do it when it is all written out, so I will show you what it looks like using the equation: Notice here that we are 4x 5 still undoing in the − 7 = -2 9 +
4 x +5 9
=5 4x + 5 = 45 4x = 40 x = 10
opposite order of what
add 7 to both sides was there. times both sides by 9 subtract 5 from both sides divide both sides by 4.
10 is the number I started with! Go ahead and make sure by sticking it into the srcinal problem, and you will see that we found the right number. We call that number a solution, because it is the only number that solves the equation.
Solving for a variable: When given a formula, it is sometimes requested that you solve that formula for a specific variable. That simply means that you are to get that variable by itself. An example: Solve for t: rt = d
(Original equation of rate x time = distance)
We are supposed to get t by itself. How do we get rid of the “r”? Divide both sides by r. It looks like this rt = d rt d = r r d t= Done. t is by itself. r
Section 3.1
94
Another example: Solve for x: y = bx +c y – c = bx subtract “c” from both sides y −c = x Divide both sides by “b”. b Done. “x” is by itself.
Section 3.1
95
Section 3.1 Exercises Part A Find the Volume of a rectangular solid when the width, height and length are given. Formula is V=lwh l = 4 in l = 7 ft l = 7.2 m 1. 2. 3.
w = 2.5 in h = 3 in V=
w = 4 ft h = 2.8 ft V=
w=9m h=3m V=
Find the Area of a trapezoid when the bases and height are given. Formula is
A=
1 2
h(B+b) B = 15 4. b = 10 h=7 A=
5.
B = 21 b = 11 h=3 A=
6.
B = 19 b=6 h = 10 A=
Simplify. 7.
2(3+x)+5(x-7)
8.
5(a-3b) – 4(a-5)
9.
3x+4y-7z+7y-3x+18z
10.
2s(t-7) – 6t(s+3)
11.
3(x -5n) +3n – 7x
12.
6kj – 7k +8kj +11
15.
-3 + m = 18
Example:
Solve.
4x + x – 7 = 1 5x – 7 = 1
Combine x’s +7 on both sides
5x = 8 8 5
x=
13.
Divide by 5 on both sides
3x − 1 + 2 = 35 7
14.
5
− 2x − 8 + 7 − 3 = 12 6
3
16.
7 3
t = 14
17.
-13 = 5x + 7
18.
5x − 6 4
19.
−
3 8
20.
12 + 2p = 3
21.
.4y = 78
x – 4 = 20
=
3
Assignment 3.1a
96
22.
5x + 3 – 7x = 15
23.
3x – 9 + 2x = - 3
24.
.3p + 5 = 19
25.
-r + 9 = -15
26.
4f + 9 = 9
27.
2x + 3 5
28.
t + t + 4t – 7 = 17
29.
=
11
5x − 8 + 7 − 3 = 18 6
3
Solve for the specified variable.
30.
y = mx + b
32.
A = 2πrh
34.
C=
5 9
for b for h
(F – 32)
for F
31.
5m − 7 3
33.
A=
1 2
bh
for b
35.
V=
2 1 π 3
for h
=
r
rh
for m
Preparation. 36. After reading some from the next section, Try to solve this problem.
Two numbers add up to 94 and the first is 26 more than the second one. Find the two numbers.
37. Find the missing variable for a cone:
r=9 l= SA = 622.04
Assignment 3.1a
97
Answers: 30 in 1.
8 5
29.
x=
2.
78.4 ft
30.
b = y – mx
or 1.6
3.
194.4 m
31.
m=
4.
87.5
32.
5.
48
33.
6.
125
34.
7.
7x – 29
35.
3r + 7 5 A h= 2πr 2A b= h F = 95 C + 32 h=
3V r2
π
8.
a – 15b + 20
9.
11y + 11z
10.
-4st – 14s – 18t
11.
-4x - 12n
12.
14kj – 7k + 11
13.
x = 12
14.
x=2
15.
m = 21
16. 17.
t=6 x = -4
18.
x=
19.
x = -64
20.
p = - 92
21.
y = 195
22.
x = -6
23.
x=
24.
p = 46.6
25.
r = 24
26.
f=0
27. 28.
x = 26 t=4
18 5
6 5
or 3.6
or 1.2
Assignment 3.1a
98
Applications of linear equations “When am I ever going to use this?” “Where would this be applicable?” All the way through math, students ask questions like these. Well, to the relief of some and the dismay of others, you have now reached the point where you will be able to do some problems that have been made out of real life situations. Most commonly, these are called, “story problems”. The four main points to remember are:
D- Data. Write down all the numbers that may be helpful. Also, note any other clues that may help you unravel the problem.
V- Variable. In all of these story problems, there is something that you don’t know, that you would like to. Pick any letter of the alphabet to represent this.
P- Plan. Story problems follow patterns. Knowing what kind of problem it is, helps you write down the equation. This section of the book is divided up so as to explain most of the different kinds of patterns.
E- Equation. Once you know how the data and variable fit together. Write an equation of what you know. Then solve it. This turns out to be the easy part. Once you have mastered the techniques in solving linear equations, then the fun begins. Linear equations are found throughout mathematics and the real world. Here is a small outline of some applications of linear equations. You will be able to solve any of these problems by the same methods that you have just mastered.
Translation The first application is when you simply translate from English into math. For example: Seven less than 3 times what number is 39? Since we don’t know what the number is, we pick a letter to represent it (you can pick what you would like to); I will pick the letter x: 3x – 7 = 39 then solve 3x = 46 x = 463 (or 15 13 or 15.3) That’s the number.
Substitution Sometimes you are given a couple of different things to find. Example: Two numbers add to 15, and the second is 7 bigger than the first. What are the two numbers?
Section 3.1 Applications: Translation
99
Pick some letters to represent what you don’t know. Pick whatever is best for you. I will choose the letter “f” for the first number and “s” for the second. I then have two equations to work with: f + s = 15 and s = f + 7 The letter “s” and “f+7” f + f + 7 = 15 are exactly the same and 2f + 7 = 15 can be changed places. 2f = 8 f=4 4 must be the first number, but we need to stick it back in to one of the srcinal equations to find out what “s” is.s = f + 7 =4+7 = 11.
4 and 11 are our two numbers.
These kind of problems often take the form of an object being cut into two pieces. Here, I will show you what I mean. Example: A man cuts a 65 inch board so that one piece is four times bigger than the other. What are the lengths of the two pieces? Now, I would personally pick “f” for first and “s” for second. We know that f + s = 65 and that s = 4f Thus,
f + 4f = 65 5f = 65 f =and 13, 52in. so the other piece must be 52. The pieces are 13in
Shapes With many of the problems that you will have, pictures and shapes will play a very important role. When you encounter problems that use rectangles, triangles, circles or any other shape, I would suggest a few things: 1. Read the problem 2. READ the problem again. 3. READ THE PROBLEM one more time.
Once you draw a picture to model the problem – read the problem again to make sure that your picture fits. The formulas for the shapes that we will be discussion are found in Section 2.2.
Sections 3.1 Applications: Substitution/ Shapes
100
Variable on Both Sides Unfortunately, not all equations come out such that this un-doing technique works. Sometimes the x shows up in several different places at once: 3x – 5 +2x – 3 = 5x + 7(x – 8)
Seeing all of the x’s scattered throughout the equation sometimes looks daunting, but it isn’t as bad as all that. We know a couple of ways to make it look a bit more simple. 3x – 5 +2x – 3 = 4x + 7(x – 8) becomes 5x – 8 = 4x + 7x – 56 5x – 8 = 11x – 56
Now we reach a point where you should feel somewhat powerful. Remember that you can add, subtract, multiply or divide anything you want! (As long as you do it to both sides). Particularly, I don’t like the way that 11x is on the left hand side. I choose to get rid of it! So, I subtract 11x from both sides of the equation: 5x – 8 = 11x – 56 -11x -11x Upon combining the like-terms, I get
Distribute the 7 and combine Combine the like terms
You might as well know that if you didn’t like the 5x on the right hand side, you could get rid of that instead:
5x – 8 = 11x – 56 -5x -5x Combining like terms, we get: -8 = 6x – 56 48 = 6x (add 56 to both sides) = x always(divide by 6) We8will get theboth samesides answer! You can’t mess up!
-6x – 8 = -56 Which now is able to be un-done easily: -6x = -48 x=8
(add 8 to both sides) (divide both sides by -6)
Special cases: What about 2x + 1 = 2x + 1 Well if we want to get the x’s together we had better get rid of the 2x on one side. So we subtract 2x from both sides like this: 2x + 1 = 2x + 1 -2x -2x Section 3.1
101
0=0 5=5 -3 = -3 solution is all real numbers
1=1 Ahh! The x’s all vanished. Well, what do you think about that? This statement is always true no matter what x is. That is the point. x can be any number it wants to be and the statement will be true. All numbers are solutions.
On the other hand try to solve: 0=1 5=7 -3 = 2 No solution
2x + 1 = 2x -2x -2x- 5 1 = -5 Again, the x’s all vanished. This time it left an equation that is never true. No matter what x we stick in, we will never get 1 to equal -5. It simply will never work. No solution.
Section 3.1
102
Section 3.1 Exercises Part B Simplify. 1.
4s(t-9) –t(s+11)
2.
12(x -5n) +3n – 4x
3.
6nj – 7j +8nj +11n
6.
-3 – 7m = 18
9.
2x − 7 3
Solve. 4.
6x − 4
5
7.
10.
7 2
+
5
2 = 30
5.
− 2x + 8
7
8.
t = -14
t +5t + 4t – 7 = 17
11.
+
6
5 − 2 = 12
-15 = 3x + 9
=
33
5x − 8 + 7 − 3 = 42 6
9
Solve for the specified variable. 12.
y = mx + b
14.
6 = 7b – pb
16.
P = 2l+ 2w
for x for b for l
13.
5m + 9 2
15.
3t + nt = y
17.
SA = 2πrh+2πr
=
r
for m for t for h
18. 27 is 6 more than 3 times a number. What is the number? 19. 18 less than 5 times a number is 52. What is the number? 20. Two numbers add to 37 and the second is 9 bigger than the first. What are the two numbers? 21. Two numbers add to 238 and the first is 34 bigger than the second. What are the two
numbers? 15in 22. Find the area of the shaded region:
8in
Assignment 3.1b
103
23.
I have created a triangular garden such thatthe largest side is 6ftless than twice the smallest and the medium side is 5ft larger than the smallest side. If the total perimeter of the garden is 47ft, what are the lengths of the three sides? 24. If
a rectangle’s length is 5 more than twice the width and the perimeter is 46 mm, what are the dimensions or the rectangle? 2
25. If
a cone has a Lateral Surface Area of 250 in, a radius of 8in, what is the slant height of the cone? Use a calculator.
26. Two
numbers add to 589 and the first is 193 bigger than the second. What are the two numbers? 27. If
3
a cylinder has a volume of 538 cm and a radius of 6 cm, how tall is it?
28.
Find the missing variable for a rectangle: P = 39 ft w = 7.2 ft l=
29.
Find the missing variable for a cylinder: 2 SA = 800 in h= r = 9 in
Solve.
Example: x + 4 – 5x = 7x + 1 -4x + 4 = 7x + 1 +4x +4x 4 = 11x + 1 -1 -1 3 = 11x 3 =x 11
Combine like terms Get all x’s together byadding 4x to both sides Subtract 1 from both sides
Divide both sides by 11
30.
4p + 2 = 7p - 6
31.
-4n + 5 = n
32.
2x – 7 = x + 5
33.
x – 42 = 15x
34.
– 4(x-3) = -2x +12
35.
7x = 13 + 7x
36.
.4x – 1 = .9x + 5
37.
2(x – 4) = 3x - 14
38.
.4y +78 = 78 + .4y
Assignment 3.1b
104
Answers: 3st – 36s – 11t 1.
28.
l = 12.3 ft
2.
8x2 – 57n
29.
h = 5.15 in
3.
14nj – 7j + 11n
30.
p=
4.
x=4
31.
n=1
5.
x = 13
32.
x = 12
6.
m = -3
33.
x = -3
7.
t = -4
34.
x=0
8.
x = -8
35.
No solution
9.
x = 53
36.
x = - 12
37.
x=6
38.
All numbers
10. 11. 12. 13. 14. 15.
16.
12 5
t=
x=-
=
b= t=
or 2.67
y−b
x= m
or 2.4 4 5
m 2r − 9
5 6
7− p y
3+ n
l = p − 2w 2
17.
h=
SA − 2πr
2
2πr
18.
7
19.
14
20.
14, 23
21.
102, 136
22.
69.73 in2
23.
12, 17, 18
24.
l = 17mm; w = 6mm
25.
26.
198, 391
27.
4.76 cm
= 9.95 in
Assignment 3.1b
105
Section 3.1 Exercises Part C Solve. 1.
4.
7.
3x + 4 + 2 = 65 5
2.
t + 1 = -11
5.
5 3 7
8t +3t + 14t – 17 = -17
8.
− 2x + 8 − 3 + 17 = 20 5
3
9 = 3x +17
3.
-17 – 7m = -18
6.
5x + 7 4
=
13
5x + 8 + 9 − 3 = 18 2
7
Solve for the specified variable. 9.
11. 13.
p= fx + bn M = 5t – 3p
E= Q−
T1 T2
for f for t
for Q
10.
F
=
xf
−
xz
2
for f
12.
LSA = πrl for r
14.
3s − 4 g 7
=
c
for g
15. 48 is 9 more than 3 times a number. What is the number? 16. 18 less than 7 times a number is 80. What is the number? 17. Two numbers add to 151 and the second is 21 bigger than the first. What are the two
numbers? 18. Two numbers add to 436 and the first is 134 bigger than the second. What are the two
numbers? 19. Find the area of the shaded region:
3cm 8cm 14cm 20. I have created a triangular garden such that the largest side is 9 less than twice the smallest
and sideof is the 7 larger the smallest side. If the total perimeter of the garden is 82, whatthe aremedium the lengths threethan sides?
Assignment 3.1c
106
21. If a rectangle’s length is 7 more than 4 times the width and the perimeter is 54 what are the
dimensions or the rectangle? 22. If a cone has a volume of 338 cm3 and a radius of 6 cm, how tall is it?
23. Find the missing variable for a parallelogram:
A = 64 in2 h= b = 12.6 in
Solve.
Example: x + 4 – 5x = 7x + 1 -4x + 4 = 7x + 1 +4x +4x 4 = 11x + 1 -1 -1 3 = 11x 3 =x 11
Combine like terms Get all x’s together by adding 4x to both sides Subtract 1 from both sides
Divide both sides by 11
24.
5p + 12 = 33 – p
25.
7n + 18 = 5(n – 2)
26.
5x – 10 = 5x + 7
27.
x – 7 = 15x
28.
2x – 4(x-3) = -2x +12
29.
.07x = 13 - .12x
30.
.7(3x – 2) = 3.5x + 1
31.
.3x – 9 + 2x = 4x - 3
32.
.4y = 78 + .4y
33.
7(x – 5) – 3x = 4x – 35
34.
9x – 4(x – 3) = 15x
35.
2x – 3x + 7x = 9x +8x
Assignment 3.1c
107
Answers: x = 17 1.
28.
All real numbers
2.
x = -6
29.
68.42
3.
m=
1 7
30.
x = - 127 or -1.71
4.
t = -28
31.
x = -3.53
32.
No solution
8 3
5.
x=-
6.
x=9
33.
All numbers
7.
t=0
34.
x=
6
or 1.2
5
8. 9. 10. 11. 12.
x = -4 f=
r
x=0
x 2 F + xz
f=
M
t=
35.
p − bn
=
x 3p
+
5 LSA l
π
13.
14.
Q
=
g
=
T1
E+
T2
7c − 3s −
4
or
15.
x = 13
16.
x = 14
17.
65, 86
18.
151, 285
19.
129.9 cm
20.
21, 28, 33
21.
w = 4, l = 23
22.
h = 8.97 cm
23.
h = 5.08 in
24.
p=
25.
n = -14
26.
no solution
27.
x = - 12
7 2
3s − 7c 4
or 3.5
Assignment 3.1c
108
Percents
Section 3.2
If you scored 18 out of 25 points on a test, how well did you do. Simple division tells us that you got 72%. As a review, 18/25 = .72 If we break up the word “percent” we get “per” which means divide 72 and “cent” which means 100. Notice that .72 is really the fraction 100 . We see that when we write is as a percent instead of its numerical value, we move the decimal 2 places. Here are some more examples to make sure that we get percents: .73 = 73% .2 = 20%
Linear Equations w/ fractions
.05 % 1 = =5 100% 2.3 = 230% The next reminder, before we start doing problems, is that the word “of” often means “times”. It will be especially true as we do examples like: What is 52% of 1358? All we need to do is multiply (.52)(1358) which is 706.16 Sometimes however, it isn’t quite that easy to see what needs to be done. Here are three examples that look similar but are done very differently. Remember “what” means “x”, “is” means “=” and “of” means times. What is 15% of 243?
x = .15(243) x = 36.45
15 is what percent of 243?
15 = x=(243) .062 x 6.2% = x
15 is 243% of what?
15 = 2.43x 6.17 =x
Once we have that down, we have the ability to solve tons of problems involving sales tax, markups, and discounts. Here are two examples:
An item sells for $85.59 but is on sale at 20% off. What is the final price? .2(85.59) = 17.12 85.59 – 17.12
amount of discount subtract discount
An item sold at $530 has already been marked up 20%. What was the price before the mark-up? x + .2x = 530 srcinal + 20% of srcinal = final price 1.2x = 530 x = 441.67
$68.47 = final price
Section 3.2 Applications: Percents
109
Section 3.2 Exercises Part A 1. 2.
45 is 12 more than 3 times a number. What is the number? 25 less than 7 times a number is 108.What is the number?
3.
Two numbers add to 251 and thesecond is 41 bigger than the first. What are the two numbers? 4.
Two numbers add to 336 and the first is 124 bigger than the second. What are the two
numbers? 5.
Find the area of the shaded region: 5cm 8cm
15cm I have created a triangular garden such thatthe largest side is 8mless than twice the smallest and the medium side is 12m larger than the smallest side. If the total perimeter of the garden is 104m, what are the lengths of the three sides? 6.
7. If
a rectangle’s length is 5 more than 3 times the width and the perimeter is 58 mm what are the dimensions of the rectangle? 8. If
2
a parallelogram has an area of 258.9 cm and a base of 23.2 cm, how tall is it?
9. Find
the missing variable for a trapezoid: 2 A = 68 ft b= h= 4ft B = 21ft
Solve. 10.
7p + 13 = 33 – 4p
11.
5n + 48 = 7n – 2(n – 2)
12.
5x – 10 = 7(x – 2)
13.
3x – 7 = 12x
14.
5x – 7(x+3) = -2x -21
15.
.06x = 15 - .18x
16.
17.
.8(7m – 2) = 9.5m + 1
18.
.2q – 7 + 2q = 3q - 5
12t = 45 + .4t
Assignment 3.2a
110
19.
6(x – 5) – x = 5x – 20
22.
18 is what percent of 58?
23.
What is 87% of 54?
24.
34 is 56% of what?
25.
What is 13% of 79?
26.
119 is 8% of what?
27.
23 is what percent of 74?
28.
Original Price:$92.56 Tax: 7.3% Final Price:
29.
Original Price: Discount: 40% Final Price: $43.90
30.
Original Price: Tax: 5% Final Price: $237.50
31.
Original Price: $58.50 Discount: 30% Final Price:
20.
9x – 2(x – 3) = 15x +7
21.
5x – 13x + x = 7x +8x
32. If the population of a town grew 21% up to 15,049. What was the population last year? 33. If the price of an object dropped 25% down to $101.25, what was the srcinal price? Preparation. 34. After reading some from the next section, try to solve this equation. 13 15 2x x + = − 7 7 7 7
35. Solve. 13 15 2x x + = − 3 3 3 3
Assignment 3.2a
111
Answers: x = 11 1.
28.
$99.32
2.
x = 19
29.
$73.17
3.
105, 146
30.
$226.19
4.
106, 230
31.
$40.95
5. 6.
136.71 cm
32.
12,437
25m, 37m, 42m
33.
$135
w = 6mm, l = 23mm
7. 8.
11.16 cm
9.
13ft = b
10.
p=
11.
No solution
12.
x=2
13.
x=
14.
All numbers
15.
x = 62.5
16.
m = - 23 or - 26 39
17.
q = -2.5
Discuss together.
34. 35.
Discuss together.
20 11
−
7 9
t = 3.879
18. 19.
No solution
20.
x = - 18
21.
x=0
22.
31%
23.
46.98
24.
60.7
25.
10.27
26.
1487.5
27.
31%
Assignment 3.2a
112
Equations with Fractions The one other thing that might throw you off is when you see a bunch of fractions in the problem. Not to worry, remember that you have power to do anything you want to the equation. For example: 3 5 7x might be easier to look at if there weren’t so many fractions in the x− = 8 8 8 way. Well, get rid of them. Multiply by 8 on both sides. 5 7x (8 ) 3 which makes it become: x − ( 8 ) = ( 8) 8 8 8 3x – 5 = 7x (not bad at all) -5 = 4x - 54 = x Ta Da. Worse example: 2 x−3 − 7 4
=
5
looks scary.
You have the ability to wipe out all of the fractions. Fractions are simply statements of division. The opposite of division is multiplication – and you have the power to multiply both sides of the equation by anything you want to. The question is, what will undo a division by 7 and by 4; the answer is multiplication by 28. Here is what it looks like:
1. Simplify
2 x−3 7− 4 =5 2 x −3 ( 28) − ( 28) = 5( 28) 7 4 (4)2 – (7)(x – 3) = 140 8 – 7x + 21 = 140
(multiplying everything by 28) (28/7 = 4 and 28/4 = 7) (Distribute the -7)
-7x +29 = 140
(Combine numbers)
2. Subtract
-7x = 111
(Subtract 29 from both sides)
3. Divide
x=
−
111 7
(Not a nice looking answer, but it is
right!)
Section 3.2
113
Every problem can be boiled down to three steps:
Linear Equations 1. Simplify
1. Parentheses 2. Fractions 3. Combine like terms
2. Add/Subtract 3. Multiply/Divide
Section 3.2
114
Section 3.2 Exercises Part B 1. 35 less than 7 times a number is 98. What is the number? 2. Two numbers add to 351 and the second is 71 bigger than the first. What are the two
numbers?
Solve. 3.
7p + 12 = 33 – 4p
4.
3n + 48 = 7 – 2(n – 2)
5.
5x – 10 = 5(x – 2)
6.
3x – 7 = 15x
7.
5x – 7(x+3) = -2x +12
8.
.09x = 13 - .18x
9.
.8(3x – 2) = 9.5x + 1
10.
.2x – 7 + 2x = 3x - 5
11.
12m = 70 + .4m
12.
5(x – 5) – x = 4x – 20
13.
9x – 4(x – 3) = 15x +7
14.
8x – 12x + x = 9x +8x
15.
85 is what percent of 39?
16.
85 is 54% of what?
17.
What is 19% of 2,340?
18.
What is 23% of 79?
19.
119 is 18% of what?
20.
43 is what percent of 174?
21.
Original Price:$72.56 Tax: 7.3% Final Price:
22.
Original Price: Discount: 30% Final Price: $49.70
23.
Original Price: Tax: 5% Final Price: $339.50
24.
Original Price: $55.50 Discount: 40% Final Price:
25. If the population of a town grew 31% up to 17,049. What was the population last year? 26. If the price of an object dropped 35% down to $101.25, what was the srcinal price?
Assignment 3.2b
115
Solve.
Example:
(12)
−
−
(12) (12) = x
Clear fractions bymultiplying by 12
+(12)
4(x+4) – 30 = 3x + 10 4x + 16 – 30 = 3x + 10 x – 14 = 10 x = 24
Distribute through parentheses Combine, getting x to one side Add 14 to both sides
27.
7 3
t – 5 = 19
28.
−
30.
4 5
x = 2x -
5 3
31.
3 5
33.
.9(-4x – 5) = 2.5x + 6
34.
3 8
(x – 7) = 5 + 3x
x–
2 5
(x-3) =
1 5
x +3
.0005x + .0045 = .004x
29.
2 3
32.
3 x+2 4 x −1 = 7 5
35.
x+
x–6=3+
4
7
=
1 2
x
8 − 56 x
Preparation. 36. Describe the best way to get rid of fractions in an equation.
Assignment 3.2b
116
Answers: 19 1.
2.
140, 211
3.
p=
21 11
4.
n=
−
5.
All numbers
6.
x=
−
37 5
or -7.4
7 12
28.
x = - 19 27
29.
x = 54
30.
x=
31.
no solution
32.
x=
33.
x = - 105 61
no solution
25 18
17 13
x=
9
x = 48.15
34. 35.
x=
7 75 13
9.
x = -.366
36.
Discuss together.
10.
x = -2.5
11.
m = 6.03
12.
no solution
13.
x=
14.
x=0
15.
218%
16.
157.4
17.
444.6
7. 8.
1 2
18.17
18. 19.
661.1
20.
24.7%
21.
$77.86
22.
$71.00
23.
$323.33
24.
$33.30
25.
13,015
26.
$155.77
27.
t=
72 7
Assignment 3.2b
117
Summary of Linear Equations
Linear Equations 1 – Simplify
Parentheses Fractions Combine like terms
2 – Add/Subtract 3 – Multiply/Divide
Word Problems D,V,P,E
Section 3.2
118
Section 3.2 Exercises Part C Solve. 1.
4.
3x − 1 − 2 = 70 5
2.
t = -24
5.
5 6 7
− 6x + 4 + 3 − 5 = 19 2
3
19 = 3x -7
3.
-4 – 9m = -22
6.
5x − 7 3
= −
9
Solve for the specified variable. 7.
9.
2s − at 2 2t d
=
=
LR2 R2
+
8.
V for s
R1
for R
10. 1
r=
I pt
9s − 5 g 11
for p
=
for s
c
11. 84 is 6 more than 3 times a number. What is the number? 12. Two numbers add to 438 and the first is 74 bigger than the second. What are the two
numbers? 14in 13. Find the area of the shaded region:
9in
14. If a rectangle’s length is 7 more than 4 times the width and the perimeter is 194 mm, what are
the dimensions or the rectangle?
15. Find the missing variable for a rectangle:
P = 48.3 ft w = 7.2 ft l=
16. Find the missing variable for a cone:
SA = 628.32 in2 r = 8 in l=
Solve. 17.
7p + 12 = 13 – 7p
18.
4n + 68 = 7 – 2(n – 2)
19.
7x – 10 = 5(x – 2)
Assignment 3.2c
119
20.
9x – 4 = 15x
23.
14 is what percent of 68?
24.
What is 37% of 754?
25.
119 is 18% of what?
26.
27 is what percent of 74?
27.
Original Price:$192.56 Tax: 7.3% Final Price:
28.
Original Price: Discount: 35% Final Price: $43.90
21.
8x – 7(x+3) = x – 21
22.
.18x = 13 - .20x
29. If the price of a meal after a 20% tip was $28.80? What was the price of the meal before the
tip was added? 30. If the price of an object dropped 15% down to $59.50, what was the srcinal price?
Solve.
31.
7
t – 2 = 19 + 5t
32.
3
34.
5 2
−
3
(x – 4) = 5 + 2x
33.
4
(-4x – 2) =
3 4
x+6
35.
5 x +8 x −5 = 3 6
1
x–4=3+
6
36.
3
x
10
x +7 = 14
6 − 73 x
Assignment 3.2c
120
Answers: x = 27 1.
28.
$67.54
2.
x = -1
29.
$24
3.
m=2
30.
$70
4.
t = -28
31.
t = - 638
32.
x = - 118
33.
x = -52.5
26 3
5.
x=
6.
x = -4
7. 8. 9. 10.
+
s = 2Vt p= R1
=
s=
2
I rt LR 2
x = - 44
2
at
−
34.
dR2
43
35.
x = -6
36.
x = 11
d 11c + 5 g
9
11.
26
12.
182, 256
13.
62.38 in
14.
18mm X 79mm
15.
l = 16.95 ft
16. 17.
17 in p = 141
18.
n = -9.5
19.
x=0
20.
x = - 23
21.
All numbers
22.
x = 34.21
23.
20.6%
24.
278.98
25.
661.1
26.
36.5%
27.
$206.62
Assignment 3.2c
121
Section 3.3
The rules that come with exponents are relatively easy to understand, but they take some practice to ensure that you have them down completely. Instead of numbers we will use letters. If we multiply: x5x8 We just have to remember what that means:
Exponents
(xxxxx)(xxxxxxxx), which is simply 13 of them multiplied together. 13
multiplication.
We write it as x .This is our very first rule! Exponents add during
x5x8=x13. The next one is quite similar: (x5)8 Again, we just have to remember what it means: (x5)(x5)(x5)(x5)(x5)(x5)(x5)(x5) which is by the first rule: x 40. That gives us the second rule: Exponent to exponent will multiply. (x5)8 = x40. Division with exponents is just about as easy. Looking at:
This means:
and we are left with x 3.
Third rule: Exponents subtract during division. This particular rule gives rise to a couple of interesting facts. Specifically, what happens if the top and the bottom have the same power?
= x0 But, we know that anything divided by itself is equal to 1. Thus: 0 x =1 Secondly, what happens if the number on the bottom is larger than the one on the top. For example:
By using the third rule we get
= =
x-3, - a negative exponent! What do we do with that?
Well, if we do it the long way, we get:
which is
Section 3.3
122
Thus we have our next definition. A negative exponent puts the number on the bottom. x-3 =
Look at a couple of examples: Using rules of exponents 23 · 22 = 25 = 2 · 2 · 2 2· 2= ·32
2
=2 =2 =8
Checking with numbers 3 2 · 2 = 8 · 4 = 32
= = 8
Look at that. The rules really work for any number. Here are some more examples to be able to simplify some expressions: (3x5)3 = 27x15 by use of the second rule. (4y5)(7y12) = 28y17 by use of the first rule.
= = = = 49 2-6 =
by the definition of a negative exponent
6-2 =
= = = = =
Here is a summary of how you can simplify expressions with exponents:
Rule
Multiplication –
add exponents Exponent to a power – multiply exponents Division – subtract
exponents Exponent of 0
Negative exponent
Official
Example
Why
aman = am+n
3x2 · 2x5 = 6x7
3xx2xxxxx = 6xxxxxxx = 6x7
(am)n = amn
(5x2)3 = 125x6
(5x2) (5x2) (5x2) = 125x6
=
364 = 9
a0 = 1 if a ≠ 0
a-n =
70 = 1; x0 = 1
1= =
2-4 =
;
364 = 9 = 9 = 1 = = =
1=
by division rule
Section 3.3
123
Section 3.3 Exercises Part A Simplify the following. (3m ) 1. 3. 5.
3-4
2.
x x11
4.
tt
6.
3x
8.
17
·
9.
(g )-
10.
11.
(2m n g )
12.
5x
7.
·
4x
4x
13. 25 less than 7 times a number is 73. What is the number? 14. Two numbers add to 251 and the second is 41 bigger than the first. What are the two
numbers?
Solve. 15.
5p + 12 = 39 – 4p
16.
5n + 48 = 7n – 2(n – 2)
17.
15x – 10 = 5(x – 2)
18.
2(x – 5) – x = 4x – 7
19.
9m – 3(m – 3) = 15m +7
20.
8x – 12x + 3 = 9x +8x
21.
45 is what percent of 39?
22.
85 is 24% of what?
23.
What is 19% of 3,517?
24.
What is 23% of 49?
25.
Original Price: Tax: 5% Final Price: $239.40
26.
Original Price: $55.50 Discount: 23% Final Price:
27. If the population of a town grew 41% up to 7,191. What was the population last year?
Assignment 3.3a
124
28. If the price of an object dropped 35% down to $11.44, what was the srcinal price?
Solve. 7 3
29.
30.
t + 5 = 19
.3(4x + 7) = 2.5x + 6 32.
−
3 8
(x + 7) = 5 + 3x
31.
.005x + .045 = .004x 33.
2 3
x–6=7+
x +7
34.
4
=
4
−
1 2
x
5 6
x
Preparation. 35. Simplify the following (so that there are no negative exponents).
Assignment 3.3a
125
Answers: 243m1 1.
28.
$17.60
2.
x
29.
t=6
3.
f
30.
4.
t1
31. 32.
x = -3
6.
x=x = 78
12x
33.
x = -45
7. 8.
1
34. 35.
x=
11.
12.
20x
13.
14
14.
105,146
15.
p=3
16.
No solution
17.
x=0
18.
x = -1
19.
m=
20. 21.
x= 115.4%
22.
354.17
23.
668.23
24.
11.27
25.
$228
26.
$42.74
27.
5,100
5.
9. 10.
5 7
128m14n g
Assignment 3.3a
126
Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly, monthly, or daily in some cases. Excel will allow you to make these calculations by adjusting the interest rate and the number of periods to be compounded. Remember that all interest rates provided in the problems are annual rates. You must adjust them to fit other compounding periods. The adjusted rate is called the periodic rate. To adjust the periodic rate in Excel, open the FV calculation box and change a 10% annual rate to quarterly, monthly, or daily as follows: Quarterly Monthly Daily
Rate: .10/4 Changing the rate to 2.5% or .025 Rate: .10/12 Changing the rate to .83% or .0083 Rate: .10/365 Changing the rate to .0274% or .000274
Change ten years of compounding to quarterly, monthly, or daily as follows: Quarerly Monthly Daily
Nper: 10*4 Changing the compounding periods to 40 Nper: 10*12 Changing the compounding periods to 120 Nper: 10*365 Changing the compounding periods to 3,650
If you assume you put $50 into savings and you are comparing savings accounts where the 10% annual interest rate is compounding quarterly, monthly, or daily. You can compare the amount of interest you will earn using Excel as follows: Quarterly Rate: .1/4 or .025 Nper: 10*4 or 40 Pmt: 0 Pv: -50
Rate: Nper: Pmt: Pv:
Monthly
Future Value = $134.25
Future Value = $135.35
.1/12 or .00833 10*12 or 120 0 -50
Daily Rate: .1/365or .000274 Nper: 10*365 or 3650 Pmt: 0 Pv: -50
Future Value = $135.90
The more frequently interest is added to your savings and compounded, the more interest you will earn. The above illustration involves a small amount of savings. The more the savings
and the more often you add to your savings the more difference it will make when the interest in added and compounded more frequently. The following example illustrates saving $100 per month for ten years at 10% interest rate compounded monthly versus annually. Annually Rate: .1 or 10% Nper: 10 Pmt: -1200 Pv: 0
Monthly Rate: .1/12 or .00833 Nper: 10*12 Pmt: -100 Pv: 0
Future Value = $19,124.91
Future Value = $20,484.50
127
= +
Savings Plan Formula for a lump sum
− + =
Savings Plan Formula with payment
Thus we have the monster formula for a Savings Plan that begins with a balance and then is added to by a payment:
A = Final Amount PMT = monthly payment P = Principal amount
(beginning balance) r = annual interest rate n = number of compounding per year Y = number of years So, = periodic interest rate (rate used in spreadsheet)
nY = number of periods (nper)
− + = + +
Spreadsheets normally have this formula built into their functions. It is known as Future Value (FV), so you won’t need to use this one if you learn the spreadsheet well.
Loan Payment Formula
= −+
Spreadsheets also normally have this formula built into their functions. It is known as Payment (PMT). Final note using a spreadsheet: The formulas are built so that money going out from you is negative and money coming in to you is positive. When you are entering Savings into the spreadsheet, the payment and Principal (Present Value) will be negative. However, for a loan, the payment will be negative but the Principal (Present Value) will be positive, because it represents money coming to you.
Section 3.3
128
Calculating Payments, Interest Rates, and Number of Periods Excel will help you calculate the payment you will need to make on a loan. It will calculate the interest rate you would need to earn on your savings to realize a certain future balance. The number of periods it will take to have your savings grow to a certain future balance can also be determined.
Monthly Payment Calculation
If you wanted to buy a car that costs $15,000 and you can get a loan at 6% interest for four years, you can determine the monthly payments using the PMT Excel function as follows: Rate: Nper: Pv: Fv:
.06/12 or .005 (monthly interest) 4*12 or 48 (months) -15000 0 Monthly Payment = $352.28
When you have paid the monthly payment for forty-eight months you will own the car and the future value of the loan is zero because the loan in paid off.
Benefits Versus Bondage You can see how hard your savings will work for you given an interest rate and enough time. However, interest works against you when you borrow money. The benefits may seem great at the moment but the financial bondage is terrible. By calculating the interest you would pay on a loan to borrow a car and the interest you would earn by saving to be able to pay cash for the car, we can determine the financial advantage of collecting interest rather than paying interest. Interest Paid on a Car Loan
You calculate the amount of interest you would pay on a four year car loan of $15,000 at 6% annual interest using the Excel Pmt function as follows: Rate: .06/12 Nper: 4*12 Pv: Fv:
-15000 0 Monthly Payment = $352.28 Total Payment = $352.28*48 (Payments) = $16,909.22 Section 3.3
129
Interest Paid =$16,909.22 (Paid) -$15,000 (Borrowed) = $1,909.22 TIP: You can have Excel calculate this for you by entering the Pmt function to calculate
the monthly payment and then, on the formula bar at the top of the Excel sheet, multiply by 48 payments and subtract the $15,000 you borrowed. The formula will be as follows: =PMT(0.06/12,4*12,-15000,0)*48-15000 You can also double click on the cell with the Pmt calculation in it and the formula will appear in the cell. Now you can multiply by 48 payments and subtract 15000 and enter this formula formula bar. in the cell. The cell will have the answer and the formula will be in the
The interest you will earn on your savings of $350.00 per month earning 6% annual interest for 39 months (the number of months we calculated above would be required to accumulate $15,000 in savings) is calculated using the FV function in Excel as follows: Rate: Nper: Pmt: Pv:
.06/12 39 -350 0 FV = $15,030.44 Amount Deposited in Savings = $350*39 (deposits) = $13,650.00 Interest Earned on Savings = $15,030.44-$13,650.00 = $1,380.44
Again, you can double click on the cell containing the FV calculation and subtract 350*39 and enter this formula giving you the amount of interest earned. You can make the same adjustment to the formula in the formula bar. The resulting formula is as follows: =FV(0.06/12,39,-350)-350*39 Total Savings From Saving Versus Borrowing
Here is how you benefited by saving and paying cash for the car rather than borrowing the money to buy the car: Interest Earned Interest Not Paid Financial Advantage
$1,380.44 $1,909.22 $3,289.66
You are wealthier by $3,289.66 because you collected interest rather than paying interest. This practice will make a major difference in your financial well being throughout your Section 3.3
130
life. If you put the money you save by paying cash for major purchases to work for you by investing it for your retirement you will add greatly to your independent wealth. You can estimate that using the FV function in Excel as follows assuming a 6% return on your investment for 30 years: Rate: Nper Pv:
.06 30 -3289.66 FV = $18,894.13
This addition to will yourmake wealth alongimpact with theonother resulting from saving rather than borrowing a major youradditions ultimate wealth. TIP: In all of the Excel functions you will be using, you only need three entries or factors to
calculate the fourth factor you are after. Notice that there are only three entries in each of the above Excel functions. You can leave blank any factor not needed and Excel will assume it is zero.
Section 3.3
131
Section 3.3 Exercises Part B Simplify the following. (3m2)3(2m2)3 1.
2.
(x7x11)3
3.
4.
t8m5t5m3
5.
2-4
6.
3x7 (4x2 – 5x +3)
7.
8.
9.
(5p-5g8)-2
10.
11.
12.
5x5 (4x7 – 7x6 + 5x-2)
13. Why doesn’t a negative exponent make theanswer negative?
Using your calculator and the Savings Plan formulas, fill out the table for a savings account. 14. Simple n = 1 15. Quarterly n = 4 16. Monthly n = 12 17. Daily n = 365 P = 200 P = 200 P = 200 P = 200 r ==8% Y 15 A=
r ==8% Y 15 A=
r ==8% Y 15 A=
r ==8% Y 15 A=
Using your calculator and the Savings Plan formulas, fill out the table for a savings account. 18. Simple n = 1 19. Quarterly n = 4 20. Monthly n = 12 21. Daily n = 365 P = 300 P = 300 P = 300 P = 300 r = 7% r = 7% r = 7% r = 7% Y = 15 Y = 15 Y = 15 Y = 15 A= A= A= A=
Using a spreadsheet and the Future Value (FV) formula, fill out the table for a savings account. Put your results in a spreadsheet called “Savings and Loan Practice.” 22. Simple n = 1 23. Quarterly n = 4 24. Monthly n = 12 25. Daily n = 365 P = 200 r = 7% Y = 15 A=
P = 200 r = 7% Y = 15 A=
P = 200 r = 7% Y = 15 A=
P = 200 r = 7% Y = 15 A=
Assignment 3.3b
132
Using a spreadsheet and the Future Value (FV) formula, fill out the table for a savings account. Put your results in a spreadsheet called “Savings and Loan Practice.” 26. Simple n = 1 27. Quarterly n = 4 28. Monthly n = 12 29. Daily n = 365 P = 300 P = 300 P = 300 P = 300 r = 8% r = 8% r = 8% r = 8% Y = 15 Y = 15 Y = 15 Y = 15 A= A= A= A=
Using your calculator, find the monthly (n = 12) payment 30. 31. 32. P = 300 P = 3000 P = 1500 r = 8% r = 9% r = 15% Y=2 Y=5 Y = 12 PMT = PMT = PMT=
for the following loans. 33. P = 23,000 r = 8% Y = 30 PMT =
Using a spreadsheet and the Payment (PMT) formula, find the monthly (n = 12) payment for the following loans. Put your results in a spreadsheet called “Savings and Loan Practice.” 34. 35. 36. 37. P = 300 P = 3000 P = 1500 P = 23,000 r = 8% r = 9% r = 15% r = 8% Y=2 Y=5 Y = 12 Y = 30 PMT = PMT = PMT= PMT = Using a spreadsheet and the Payment (PMT) formula, find the monthly (n = 12) payment for the following loans. Put your results in a spreadsheet called “Savings and Loan Practice.” 38. 39. 40. 41. P = 500 P = 4800 P = 2500 P = 23,000 r = 4% r = 9% r = 15% r = 8% Y=2 Y=5 Y = 12 Y = 20 PMT = PMT = PMT= PMT =
42. Ensure that every member of the group is able to putin the formulas and use the spreadsheet to do the calculations.
Assignment 3.3b
133
Answers: 216m12 1.
2. 3. 4.
x54 t13m8
28.
992.08
29.
995.90
30.
13.57
31.
62.28
32.
22.51
33.
168.77
34. 35.
13.57 62.28
36.
22.51
11.
12x9 – 15x8 + 9x7 1
12.
20x12 – 35x11 + 25x3
39. 99.64
13. 14.
Negative exponents mean division 634.43
41.
192.38
15.
656.21
42.
Complete only when everyone understands and can enter the formulas on their own.
16.
661.38
17. 18.
663.94 827.71
19.
849.54
20.
854.68
21.
857.21
22.
551.81
23.
566.36
24.
569.79
25.
571.47
26.
951.65
27.
984.31
5. 6. 7. 8. 9. 10.
37. 168.77 38. 21.71
40.
37.52
Assignment 3.3b
134
Section 3.3 Exercises Part C – Exam Review Solve. 1.
4.
3x − 1 − 2 = 10 5
2.
t = -48
5.
5 6 7
− 6x + 4 + 3 − 5 = 25 2
3
19 = 7x - 39
3.
-7 – 9m = -22
6.
4x − 7 3
= −
9
Solve for the specified variable. 2
7. 9.
2s + at 5t
d
=
=
V for s
LR2 R2
+
R1
for L
8.
r= I pt
10.
9s − 5 g 11
for t
=
for g
c
11. 84 is 6 more than 13 times a number. What is the number? 12. Two numbers add to 438 and the first is 72 bigger than the second. What are the two numbers? 14in 13. Find the area of the shaded region: 6in
14. If a rectangle’s length is 5 more than 4 times the width and the perimeter is 180 mm, what are the dimensions of the rectangle?
15. Find the missing variable for a rectangle: P = 78.3 ft w = 17.2 ft l=
16. Find the missing variable for a cylinder: 2 SA = 453.9 in r=7 h= Solve. 17.
7p + 12 = 15 – 7p
18.
3n + 68 = 7 – 2(n – 2)
19.
2x – 10 = 5(x – 4)
Assignment 3.3c
135
20.
18 is what percent of 68?
21.
119 is 28% of what?
22.
Original Price:$ 92.56 Tax: 7.3% Final Price:
23.
Original Price: Discount: 35% Final Price: $13.90
24. If the price of a meal after a 20% tip was $16.08? What was the price of the meal before the
tip was added?
25. If the price of an object dropped 15% down to $413.10, what was the srcinal price?
Solve.
26.
5 2
(-3x+ 2) =
3 4
x+6
27.
x −5 4 x +8 3 = 6
28.
2 x +7 21 =
6 − 73 x
29. Find the price, interest rate and years of a loan for homes in your area. In your “Life Plan”
spreadsheet, enter the Price, Number of years, and Interest Rate, then use the PMT formula to figure out how much it will cost to own a home. Report to your group when you have completed it. 30. Using the PMT formula in your “Life Plan” spreadsheet, find the cost of owning your own
transportation. Report results to your group. 31. Create a Visual Chart on one side of a piece of paper for Chapter 3 material including information and examples relating to Linear Equations and Applications.
Assignment 3.3c
136
Answers: x=7 1.
28.
2.
29.
x= Will be submitted in Portfolio
30.
Will be submitted in Portfolio
31.
Make it nice.
3. 4. 5. 6.
x=-
t = -56 x= x = -5
11.
== =
12.
255, 183
13.
55.73in
14.
l = 73mm, w = 17mm
15.
21.95 ft
16.
3.32 in
17.
p=
18. 19.
n=-
7. 8. 9. 10.
6
20.
x= 26.5%
21.
425
22.
$99.32
23.
$21.38
24.
$13.40
25.
$486.00
26.
x=x = -9
27.
Assignment 3.3c
137
Chapter 4 CHARTS,LINES GRAPHS, and
Overview lgebra 4.1 Charts and Maps 4.2 Lines and Slope 4.3 Writing Equations of Lines
138
Section 4.1 Graphs and Charts
Have you ever had difficulty finding locations of objects on maps? If you haven’t yet had that experience, then I have a little activity for you. At the end of your Bible (King James Version – LDS Edition) there are several maps. On any of the 13 or so maps, try to find the following locations: Bethsaida Samothrace Iconium Kir-hareseth Unless you have some help, it might take you a while. Let’s walk through a couple of them together. Right before the maps is an Index of Place-Names. First, we look up Bethsaida. In the edition I have, I find Bethsaida and right next to it is listed 11:C3. The map we have to look at is number 11, but what does the C3 mean? Well, if you turn to map #11 you will notice that across the top are letters and then there are numbers along the side. If you go straight down from C and straight across from 3, you will be right in the vicinity of Bethsaida (right on the north shore of the Sea of Galilee. Next we will look at the Samothrace. In the Index of Place-Names we find that it is located on map 13: E1. Go to map #13. Again, the letters are across the top and numbers are listed on the side. Go straight down from E and across from 1, and you will find a small island with the name of Samothrace. Here is an example from the maps at the end of the Doctrine and Covenants: Find Harmony, Pennsylvania. In the Index of Place-Names we see that Harmony, Pennsylvania is on two maps, 1:B3 and 3:H3. So we go to map #3 straight down from H and straight across from 3 and find Village of Harmony. If you aren’t familiar with the map, the little grid set up by using one letter and one number is absolutely indispensible. Because letters and numbers go in definite orders, they are called ordinates. When they are used together to pinpoint an exact location, they are then called coordinates. The use of coordinates to find an exactly location was introduced into mathematics centuries ago by a man named Rene Descartes. Now his method to specify locations is used widely in the world. Longitude and latitude are the two numbers that, when used together, can give us an exact location on the planet and form the basis for all ship and plane navigation.
Using coordinates is also valuable in being able to read charts and “see” trends that aren’t so readily picked up by only seeing the numbers. Here is an example of a savings account and how it has grown:
139
Year Amount 2001 $8.31 2002 $17.48 2003 $28.00 2004 $56.39 2005 $72.48 2006 $85.34 The chart helps to visualize the growth. Notice how the graph is made by plotting each set of coordinates. The standard coordinate system that is used in math is called the Cartesian Coordinate System. It was created by Descartes (hence the name Cartesian) and uses numbers (positive or negative) for both the horizontal and the vertical measuring. Here is the system and the parts of it:
y - axis tick marks so you won’t lose your place
x - axis
Keeping things in order, when you are given a set of numbers like (6,-2), we have the following: Parentheses tell us that these two numbers go together to make a single set of coordinates 6 comes first and so matches up with the x -2 comes second and matches up with y So, we go to where we are at 6 on the x and -2 on the y to find the right location of the point like this:
·
(6,-2) Section 4.1
140
Section 4.1 Exercises Part A Find the following locations in the maps section of your Bible using the Index of PlaceNames. 1. Marah 2. Haran 3. Mt. Ararat 4. Golgatha Now use the Church History maps. 5. Adam-Ondi-Ahman 6. Nauvoo, Illinois
Based on the chart:
7. In what year did the
company make $450? 8. How much did the
company make in 2006? 9. What years did the
company make over $500?
10. Chart the following table of growth of a savings account:
# of years Amount
0
1
2
3
4
5
6
7
8
38
50
72
105
130
155
195
205
170
Graph the following points on a Cartesian coordinate system. 11. (1,8) 12. (2,-4) 13. (3,8) 14. (-5,1) 15. (-3,-7) 16. (0,1) 17. (-2,4) 18. (1,-1) 19. (5,0)
Assignment 4.1a
141
Answers: On maps 1.
2.
On maps
3.
On maps
4.
On maps
5.
On maps
6.
On maps 2003
7. 8.
About $350
9.
2004(maybe), 2009, 2010
10.
1119
· · · · ·· · · ·
(1,8) (3,8)
(-2,4)
(-5,1)
(0,1) (1,-1)
(5,0)
(2,-4)
(-3,-7)
Assignment 4.1a
142
Section 4.1 Exercises Part B Find the following locations in the maps section of your Bible and Doctrine and Covenants (Church History) using the Index of Place-Names. 1. Thessalonica 2. Ur 3. Mt. Nebo 4. Kidron Valley 5. Nineveh 6. Fayette, New York Based on the chart:
7. In what years did the
company make about $400? 8. How much did the
company make in 2002? 9. What year did the
company make less than $375?
10. Chart the following table of growth of a savings account:
# of years Amount
0
1
2
3
4
5
6
7
8
238
301
314
325
394
420
447
439
480
11. As a group, use a spreadsheet to make a table of the growth of a savings account for 20 years
that begins with $200 and receives a $25 deposit each month and grows at 6%. The table should show the yearly values at the end of each year. 12. As a group, use the spreadsheet to make a graph of the 20-year table. 13. How much money was paid into the savings account over the 20 years? How much interest
was earned? Graph the following points on a Cartesian coordinate system. 14. (5,1) 15. (3,-7) 16. (0,-1) 17. (-12,4) 18. (1,1) 19. (-5,0)
Assignment 4.1b
143
Answers: On maps 1.
2.
On maps
3.
On maps
4.
On maps
5.
On maps
6.
On maps
11.
Answer on next page . . .
7. 8.
2001, 2005, and 2007 About $500
12. 13.
Answer on next page . . . $6200, 6013.06
9.
2006
10.
1419
·
(-12,4)
· ·· · · (-5,0)
(1,1) (0,-1)
(5,1)
(3,-7)
Assignment 4.1b
144
Answer to #11:
Answer to #12:
Assignment 4.1b
145
Section 4.1 Exercises Part C Find the following locations in the maps section of your Bible and Doctrine and Covenants using the Index of Place-Names. 1. Bethlehem 2. Ephesus 3. Hebron 4. Mt. Sinai 5. Liberty, Missouri 6. Kirtland, Ohio Based on the chart: 7. In what year did the value first drop
below 2500? 8. How much value was lost in the 7
years? 9. Which year had the biggest drop in
value? 10. Chart the following table of growth of a savings account:
# of years Amount
0
1
2
3
4
5
6
7
8
52
35
48
54
62
73
68
72
85
11. Use a spreadsheet to make a table of the growth of a savings account for 20 years that begins
with $200 and receives a $50 deposit each month and grows at 6%. The table should show the yearly values at the end of each year. 12.Use the spreadsheet to make a graph of the 20-year table. 13. How much money was paid into the savings account over the 20 years? How much interest
was earned? 14. How much interest is earned if the account grows at 9% instead of 6%? Graph the following points on a Cartesian coordinate system. (13,-7) (0,4) 15. (7,1) 16. 17. 18.
(8,4)
19.
(-1,1)
20.
(-8,0)
21. Create a realistic savings plan for yourself. Make a table of the growth of your savings for 20
years. Create a graph of the table. Include it in your portfolio. Assignment 4.1c
146
Answers: On maps 1.
2.
On maps
11.
Answer on next page
3.
On maps
12.
Answer on next page
4.
On maps
13.
$12200, $11564.09
5.
On maps
14.
$22,396.17 interest for a total of $34,596.17
6.
On maps
7.
3r year
8.
About $2000
9.
From the 1st to 2n year
10.
1520.
· 21.
(-8,1)
· ·
(0,4)
(-1,1)
·
·
(8,4)
(7,1)
·
(13,-7)
Include in Portfolio
Assignment 4.1c
147
Answer to number 11:
Answer to number 12:
Assignment 4.1c
148
Section 4.2
When we solved equations that looked like 3x-2=13, we got a solution like x=5, but what does that really mean? We have followed an algorithm whole to arrive at the proper location, but the reader is reminded that the purpose of manipulating equations is to find numbers for x that we can stick in and make a true statement. If we stick in 5 for x in this equation, we get 3(5) - 2 = 13, which is true. There is no other number which will do this. We call this a solution to the equation. Graphing
In that kind of equations we found a number for x that made the statement true, and sometimes we could even guess what would work without really using any formulas or steps. This process becomes a bit helpful when studying the next type of equation: 3x+2y=5 In this type of equation there is an x and a y to find numbers for. The solution to this equation will not be a single number as it was in the earlier cases, but a pair of numbers. The answers will look something like (3,-2), which means thatwe will stick in 3 for x and -2 for y. If you stick those in, the equation becomes: 3(3)+2(-2) = 5 9-4=5 Woo Hoo! It works! We found a solution, and we don’t even know what we are doing yet. Let’s see if there is another one. Try the following pairs of numbers in the equation to see if they also work: (1,1) (3,2) (-1,4) (5,-2)
3(1)+2(1)=5 3(3)+2(2) =5 3(-1)+2(4)=5 3(5)+2(-2)=5
solution nope solution nope
(5,-5) 3(5)+2(-5)=5
solution
If you try all of these, you will realize that some of them work as solutions and some of them don’t. In any case, you should be able to realize that there are a whole lot of solutions, tons of them! One way to get them is to keep guessing. When you get tired of that there is an algorithm that might makethings a little easier. If we pick a number for x and stick it in, then we will have an equation that we can solve for y. For example, if we say in this example we want x to be 7, we stick it in to get: 3(7)+2y=5 21+2y=5 2y=-16 y=-8 Which means that when x is 7, y will be -8, or in other words, the pair (7,-8) is a solution. What would we get if we madey = 9? The equation would be: 3x+2(9)=5
Solving for x, we get x= - , so the pair (-
,9) is a solution.
Now we can get so many solutions this way that it doesn’t pose a problem to find one anymore. Since there are so many, the question arises, “Are there any patterns in the solutions to these equations.” Well, of course there are. This is math! The solutions are pairs, which we can stick on a graph. If we plot the ones that we have already found to the problem we are using we get this: (3,-2), (1,1), (-1,4), (5,-5), (7,-8) Section 4.2
149
You will notice that all of the solutions line. If we connect them, we get all of the equation. It is important to realize the line that connects the dots,all of the line are solutions. The problems will graph the line 3x+2y=5 or something
.
are in a straight the solutions for that if we draw points on that simply ask you to similar.
.
The
.
correct answer to “Graph the line
3x+2y=5”, is then the graph at the left.
. For the next one, find four points on the line and then graph it: y = 14 x – 2 x y 1 when x = 4 we have y = 4 (4) – 2 which means y = -1. 4 when x = 0 we have y = 1 (0) – 2 which means y = -2. 4 0 0 when y = 0 we have 0 = 14 (x) – 2 which means x = 8. 3 when y = 3 we have 3 = 14 (x) – 2 which means x = 20. x y 4 -1 0 -2 8 0 20 3 Notice that we really only need two points to get the pattern. For convenience, we often select 0 for x, and then 0 for y. When x is 0 the point is on the y-axis. Likewise, when y is 0 the point is on the x-axis. In the previous example, the point (0,-2) lies on the y-axis and is called they-intercept; the point (8,0) lies on the x-axis and is called thex-intercept.
The table completely filled out looks like this: and the graph like this:
An x-intercept happens when y is zero, and a yintercept happens when x is zero.
Section 4.2
150
Here is another example. Graph the line 4x + 3y = 8; find the x and y intercepts. We start by finding the x- and y-intercepts with a table that looks like: Then fill it out by plugging in 0 for x and getting 4(0) + 3y = 8 3y = 8 y = 83 When we plug in 0 for y we get: 4x + 3(0) = 8 4x = 8 x=2
so we have the table:
x y 0 0
x y 0 83 2 0
and the graph:
There are a couple of particular kinds of lines that may give you a bit of trouble when you first see them. Your first reaction when asked to graph the line: x=4 is probably something like, “Hey, where is the y?” or, “How do I do that? It looks different.” Relax, these kind are actually a bit easier than the other ones. Watch: What is x when y is 7? Answer: 4 The points (4,7) (4,0) What is x when y is 0? Answer: 4 and (4,-3) are part of What is x when y is -3? Answer: 4 the line and help us graph it. Do you see how nice that is? Since y is not in the equation it can be anything it wants to be, but x is always 4. The graph is as follows: Here is the line x = 4; notice that it is vertical and hits where x is 4.
Section 4.2
151
For future reference you can remember that all equations that only have an x will be vertical. The other special case that may seem difficult at first looks like: y = -2 But I think you can see that it will be very similar to the previous example: What is y when x is 0? Answer: -2 What is y when x is 5? Answer: -2 What is y when x is -3? Answer -2
The points (0,-2) (5,-2) and (-3,-2) are part of the line and help us graph it.
See how slick that is?! The graph is as follows: Here is the line y = -2; notice that it is hits where y is -2.
horizontal and
All equations with just a y in them will lines.
be horizontal
Now that we can graph any lines, there is one particular property of lines that is most
7%
useful. We introduce this by bringing to it mind a familiar road This warns7 of steepness, but take a look at what is really saying. 7%sign. means thesign fraction 100 .The interpretation of the sign means that the road falls vertically 7 feet for every 100 feet that you travel horizontally. In this way the highway department uses fractions to denote the steepness of roads. We are going to do the same thing with the steepness of lines. When we have a couple of points on the graph we can find the steepness between them. Here are a couple of examples.
The steepness of the line between point A(-2,1) and B(3,3) is found by taking how much it changes up and down (distance C (1,7) between 1 and 3 = 2) over how much it changes 1st Example left and right(distance between -2 and 3 = 5). That makes a steepness of 52 . The name for steepness is B (3,3) slope, and the symbol is m (as in a mountain). We would A (-2,1) write that m = 2 . 5
2n Example
The slope of the line through A(-2,1) and C(1,7) would be 6 (the distance from 1 to 7)
D (5,-3)
Section 4.2
152
over 3 (the distance from -2 to 1.
We would write m = , or in other words m = 2.
r
3
The slope of the line through A(-2,1) and D(5,-3) would be -4 (the distance from 1 down to -3) over 7 (the distance from -2 to 5); m = -74 .
Example
There are some properties that you should start to see from these examples. 1. Bigger numbers for slope correspond to steeper lines. 2. Positive slopes head up as you go to the right. 3. (Opposite of #2) Negative slopes will head down as you go to the right. In the first example we obtained the 2 as the distance from 1 to 3. What operation finds distance? Answer: Subtraction. Aha! Seeing that, we can start to see a pattern in how to find slopes a little more quickly. Let’s look at those three examples, using subtraction this time: As a note: You should realize that the subtraction may happen in the opposite direction but will still give the same slope. Example #1 would look like this: 1− 3 2−3
−
=
st
n
1 Example: 3 −1 2 = 3 − −2 5
2 Example: 7 −1 6 = = 2 1 − −2 3
rd
3
Example: 4 = − 5 − −2 7 − 3 −1
Now, see if you can find the slope between two general points: Point 1 and point 2 with coordinates that we don’t know. We would like to call them both just (x,y), but then subtraction −2 = 2 would give us zero. This is a good place to introduce you to how −5 5 subscripts can be very helpful. We will call point #1 (x 1,y1) showing that st the x and the y come from the 1 point. Similarly we will call point #2 (x2,y2). Now you can find the slope just like we did in the previous examples: y − y1 m= 2 x 2 − x1
Voila! You have just created the formula for finding slope between two points. Practice using it quite a bit until it almost becomes natural. Memorize it! Sometimes formulas are written in a few different ways. Here are some of the others: m=
y1
−
y2
x1
−
x2
m=
ychange xchange
m
=
rise run
m
=
∆y ∆x
They all mean the same thing.
Section 4.2
153
Section 4.2 Exercises Part A 1. Two numbers add up to 57, and the first is 23 bigger than the second. What are the two
numbers? 2. An international phone call costs 35¢ to connect and 12¢ for every minute of the call. How
long can a person talk for $3.60? 3. A 52m rope is cut so that one piece is 18m longer than the other. What are the lengths of the
pieces? 4.
Original Price:$292.50 Discount:20% Final Price:
5.
Original Price: Discount: 40% Final Price: $73.90
6. The perimeter of a rectangle is 82 cm. If the length of the rectangle is 6 more than 4 times the
width, what are the dimensions of the rectangle?
Fill out the table for each of the following: Ex. 1 3x + 4y = 7
x 2
3(2) + 4y = 7 4y = 1 4
1 5
3(0) + 4y = 7 4y = 7 y = 74
7. x + y = 9
x 5 -4
y
3x + 4(0) = 7 3x = 7 x = 73
0
1
y=
Solution:
y
3 0 7
0
3x + 4(1) = 7 3x = 3 x=1
x 2
y
7 3
0
113 -4
1 5
0
1 4
7 4
3x + 4(5) = 7 3x = -13 x = - 133 8.
2x – y = 5 x 2 0 -1
y
0 4
9.
5x + 4y = 9 x 1 0 -3
y
0 5
10. x–7y = 13
x
y 1 3
2 0 -1
Assignment 4.2a
154
Graph the following lines, and label three points . Example: 2x – 7y = 3
Pick three numbers to make a table (intercepts are helpful): x 0 1 -2
11.
3x + y = 10
. ..
y (0,- 37 ) (1,- 17 ) (-2,-1)
12. y = 2x
13. x – 4y = 7
14. x = 3
15. y = - 37 x + 4
16. 6x – 5y = 12
17.
18. 5x + 2y = 6
y = -4
Preparation 19. After reading a bit of section 4.2, try to find the slope between (4,1) and (7,11).
Assignment 4.2a
155
Answers: 17, 40 1.
2.
27 minutes
3.
17m, 35m
4.
$234
5. 6.
$123.17
7.
8.
(2,0) (0,- ) (7,6)
11.
(0,10) (3,1) (-1,13) 16.
12.
(0,0) (1,2) (2,4)
17.
(0,-4) (2,-4) (37,-4)
13.
(7,0) (3,-1) (0,)
18.
(0,3) (2,-2) ( ,0)
14.
(3,0) (3,1) (3,2)
19.
m =
7cm X 34cm
x 5 -4 6 9 2
y 4 13 3 0 7
x 2 0 -1
y -1 -5 -7 0
4
9.
x 1 0
y 1
-3
6
0
10.
5
x y 20 1 34 3 2
15. (0,4) (7,1) (14,-2)
0 6
-1
Assignment 4.2a
156
Section 4.2 Exercises Part B Three types of trees are in a local park. The number of aspens is 4 more than twice as many oaks, and the number of maples is 50 more than the number of oaks. There are a total of 874 trees in the park. How many of each kind are there? 1.
2. If
the length is 3 more than 4 times the width of a rectangle and the perimeter is 76mm, what are the dimensions? 3. Solve.
4.
4(x-7) = 2x + 15
Original Price:$392.50 Discount:20% Final Price:
Original Price: Discount: 45% Final Price: $73.90
5.
6. If
my vehicle can get 32 miles per gallon and fuel costs $2.75 per gallon. How many miles per dollar do I get?
Fill out the table for each of the following: 7.
2x + y = 9 x 5 -4
y
3 0 7
8.
y = 5x+2 x 2 0 -1
9.
x + 4y = 9 x 1 0 -3
y
10.
x
y
y 2 5
2 0
0 5
0 4
y = x - 13
-1
Graph the following lines, and label three points.
11. 14.
3x + 2y = 10 x = -6
12.
y = 2x - 7
13.
y= x
15.
y = - 37 x - 2
16.
2x – 5y = 12
Assignment 4.2b
157
17.
y=5
18. 5x + y = 6
Find the slope between each pair of points. Ex. (7,2) (-3,5)
m=
5−2 −3−7
=-
3 10
19. (5,-2) (7,3)
20. (4,1) (-5,6)
21. (5,-1) (-3,-8)
22. (7,3) (-2,3)
23. (-5,2) (4,-3)
24. (-6,1) (-6,5)
25. Explain the difference between a slope of zero and an undefined slope.
Preparation 26. Find two points of each line and then use those points to find the slope.
2x – 3y = 1
y = x + 4
Assignment 4.2b
158
Answers: 205 Oaks, 414 Aspen, 1. 255 Maple w=7, l=31 2.
3. 4. 5. 6. 7.
(6,0) (0,- ) (1,-2)
12.
(0,-7) (1,-5) (2,-3)
17.
(0,5) (-2,5) (3,5)
13.
(0,0) (2, 1) (8,4)
18.
(0,6) (1,1) ( ,0)
14.
(-6,0) (-6,1) (-6,2)
19. 20. 21. 22. 23. 24. 25.
m = m = - m = m=0 m = - m = undefined
$134.36 11.64 miles per dollar x 5 -4 3
x 2 0 -1 -
10.
16.
1
9.
(0,5) ( ,0) (2,2)
x= $314
8.
11.
x 1 0 -3 9 -11
y -1 17 3 0 7 y 12 2 -3 0 4
y 2
3 0 5
x y 35 2 42 5 2
15. (0,-2) (7,-5) (-7,1)
Undefined is straight up and down, vertical. 0 is horizontal, straight across
26. m = m=
0 -13 28 -1
Assignment 4.2b
159
Section 4.2 Exercises Part C Three types of trees are in a local park. The number of aspens is 4 more than twice the number of birch, and there were 50 more pines than birch. There are a total of 874 trees in the park. How many of each kind are there? 1.
2. If
the length is 7 more than 4 times the width of a rectangle and the perimeter is 74mm, what are the dimensions? 3. Solve.
4.
6.
5(x-7) = x + 15
Original Price:$92.50 Discount:20% Final Price:
Original Price: Discount: 25% Final Price: $174.30
5.
What is the Volume of a Cylinder with radius 8cm and height 12cm?
Fill out the table for each of the following: 7.
2x + 3y = 9
8.
y = -5x+2
9.
x - 7y = 9
10.
y= x
x 5 -4
y
3 0 7
x 2 0 -1
x 1 0 -3
y
x
y
2 0
0 5
0 4
y 2 5
-1
Graph the following lines, and label three points.
11. 14. 17.
4x + 2y = 10 x=5 y = -3
12.
y = -2x - 7
13.
y= x
15.
y = - 37 x - 2
16.
7x – 5y = 12
18.
5x + 2y = 6
Assignment 4.2c
160
Find the slope between each pair of points. 19. (4,-2) (7,3)
20. (4,8) (-5,6)
21. (-3,-1) (-3,-8)
22. (7,7) (-2,3)
23. (-5,-3) (4,-3)
24. (-6,1) (-5,5)
25. Explain the difference one more time between a slope of zero and an undefined slope.
Find two points of each line and then use those points to find the slope 26. 2x – 3y = 1 27. y = x + 4 28. 5x – y = 10 29. 2x + 7y = 1
30. y = - x + 3
Assignment 4.2c
161
Answers: 205 Birch, 414 Aspen, 1. 255 Pine w = 6, l = 31 2. 3. 4. 5.
$232.40
6.
2412.74 cm x 5 -4 0
-6
y
16.
( ,0) (0,- ) (1,-1)
12.
(0,-7) (1,-9) (- ,0)
17.
(5,-3) (7.2,-3) (0,-3)
13.
(0,0) (2, 3) (4,6)
18.
(0,3) (2,-2) ( ,0)
14.
(5,2) (5,0) (5,-3.4)
19.
m = m = m is undefined m = m=0 m=4
-
3 0 7
8.
x 2 0 -1
y
0
-
-8 2 7
4
9.
x 1
y
0
-
-3
-
9
0 5
44
10.
(0,5) (,0) (1,3)
x= $74.00
7.
11.
x
2 0 -
20.
-
21.
22.
23.
24.
y 2 5
15.
(0,-2) (7,-5) (-7,1)
25.
Undefined is straight up and down, vertical. 0 is horizontal, straight across
26.
(0,- ) ( , 0) m =
27.
(0,4) (5,7) m =
28.
(2,0) (0, -10) m = 5
29. 30.
(0, ) (,0) m = - (0,3) (7,1) m = -
0
-1
Assignment 4.2c
162
Section 4.3 Graphing Equations with Slope
Okay, now that you know how to graph a line by getting some points, and you know how to find the slope between two points, you should be able to find the slope of a line once you have an equation:
Example: Find the slope and graph the line 3x-4y=2 Well, if we find a couple of points: (2,1) and (6,4), the graph must look like this:
(6,4)
Then finding the slope, we can just use the same method that we have done the other ones we get the slope 3 4 1 m . 6 2 4 =
−
−
(2,1)
=
Trying this a couple of times on various equations, you might get something like the following:
Equation 3x – 5y = 10
Slope: m= 35
I hope that you kind of see a pattern emerging that you
would be able to use as a shortcut. Do you see how the change in y is always the coefficient of x? And do you see that the change in x is always the opposite of the 5x + y = 15 m = -5 coefficient of y? x-3y = 12 m = 13 These equations are all written the same way and have the same pattern for getting the slope without actually figuring it out from a couple of points. 2x + 9y = 4
m = - 29
Here is a pattern for another common way of writing lines. Pick out the pattern here: Equation: y = -2x – 5 y = 37 x + 4 4
y = - 9 x – 13 y = 7x - 2
Slope: m = -2 m = 73
The pattern here is even easier than the first one.When y is by itself, the slope is simply the number in front of x. No change.
4
m = -9 m = 17
Section 4.3
163
Since these are two very common ways of writing lines, they need some comparison. Standard Form: Slope-intercept Form: - It is written in the form Ax+By = C where - Written in the form y = mx+b. A, B, and C are integers (usually). - m is the slope without any adjustment. - Intercepts are found by putting in 0 for - (0,b) is the y-intercept. either x or y so each is relatively easily found as (0, CB ) and ( CA ,0). -
Slope is always m = - BA
Advantages: It has no fractions. Both x- and y- intercept have same amount of calculation Disadvantages: Minor calculation for y-intercept. Remembering to put the negative sign on the slope.
Advantages: Slope is most easily found. Y-intercept is most easily found. Prepares you for function f(x) notation. Disadvantages: Fractions are often part of the equation.
Since both of them will be given to you to graph, you should be able to work with both of them. Important Note: You should also see that we can change from Standard form into Slope-intercept form (and vice-versa) simply by solving for y. In the example 3x-4y=2, we get: 3x-4y=2 -4y=-3x+2 Standard: 3x-4y=2 Slope-intercept: y=34 x - 12 y= 34 x - 12 3 Slope: m= 34 Slope: m= 4 y-intercept: (0, - 12 ) y-intercept: (0, - 12 ) They are simply two different ways to x-intercept: ( 23 ,0) x-intercept: ( 23 ,0) write the same equation. There is no difference, except that of convenience. The first way is called standard form, and the second is called slope-intercept form. Again, they are the same line! Every point that works in one will work in the other.
In any case, you will learn and have practice with both forms. Being able to pick out intercepts and slope from lines will help you to graph them quickly. Having the slope especially makes it a cinch to graph lines. You only need to find one point, then follow the slope to the next point and draw the line. Example: Graph the line and find the slope of y=- 53 x - 4
-5 3
Section 4.3
164
Well the slope is rightin front of x, so m= -53 One easy point is to stick in zero for x. We get the point (0,-4). Following the slope, (it is negative, so we will head down as we go to the right) down 5 over 3 and we come to the point (3,-9), and then draw the line.
Another example: Graph the line and find the slope of 2x-7y=4 Well the slope is the opposite of 2 over -7, so m=-27 = 72
7
−
It appears that the easiest point in this one is the x-intercept, so we stick in zero for y and get x=2: (2,0). Following the slope we move up 2 and over 7 to the next point (9,2), and then draw the line.
2
That covers graphing and finding the slope for the vast majority of equations. As you will recall, there were a couple of special cases where either the x or the y were missing. We now look to find the slope of these. We will work two examples of this: First: y= -2 Remember how to find a couple of points that work: (3,-2) and (-1,-2). It gives us the graph of a horizontal line where y is always -2: Putting those two points in to the formula for finding slope, we get: 2 2 0 m= 0 1 3 4 which means that all horizontal lines will have a slope of 0. −
− −
=
−
−
=
−
Second: x=5 Remember how to find a couple of points that work: (5,2) and (5,6). It gives us the graph of a vertical line where x is always 5:
Now if we put the points in the slope formula, we get: 6 2 4 = = bad news. (Division by zero is −
−
5 5 0 undefined.) which means that all vertical lines have undefined slope.
Section 4.3
165
m = undefined m
=
15 m
=
To get a feel for slope a little bit better, we are going to take a little time to look at some slopes. You will notice that the higher the number, the steeper it is. Common m 1 sense from that will tell you that a slope of 0 will belong to a line that is completely flat. Also, you should see that since numbers get bigger as 1 the slope gets steeper, the slope of a vertical m
2
=
=
m
m
2
=
=
0
m
m
m m m
=
8
=
=
1
−
1 3
= −
line have to benumbers far greater a billion. Onwould the other hand, get than increasingly large in the negative direction for lines that are heading down ever steeper. That means that vertical lines would have to have a slope that is less than negative one billion. Hmmmmmm…. greater than a billion and less than negative a billion at the same 1 time. No wonder that division by zero 5 can’t be done and is undefined.
2 3
A word of caution: Since the term “no slope” is interpreted by some to mean zero slope and by others to mean that the slope doesn’t exist, we will simply avoid the term. A vertical line has undefined
= −
2
−
slope and a horizontal line has a slope of zero.
−
m = undefined
Section 4.3
166
Now that we can go from the equation of a line to the finding of points, getting the slope and graphing the line, we are going to work on how to go backwards. It really isn’t as difficult as it seems. Since from an equation we can get the slope, we can certainly write an equation from the slope. Example: Write an equation of the line that has slope m= 35 , and goes through the point (5,2). There are two ways to do this. Remember with standard form, the slope is the negative of the first number over the second. In slope-intercept, the slope is right in front of the x when y is by itself. Standard Since the slope is
Again, please note that if you take the standard form and solve for y, you will
Slope-Intercept 3 5
, we know that an
equation would have to be: 3x – 5y = something but what? Ahhh, here is where we use the fact that (5,2) has to work in the equation. If we stick in 5 for x and 2 for y we get: 3(5) – 5(2) = 15 – 10 =5 3x – 5y = 5
Since the slope is
3 5
, we know that an
equation would have to be: y = 35 x+b but what is b? Ahhh, here is where we use the fact that (5,2) has to work in the equation. If we stick in 5 for x and 2 for y we get: 2 = 35 (5) + b 2=3+b -1 = b Thus our equation must be y=3x - 1
get the slope-intercept form.
5
As a side note on slope: When two lines have the same slope, or steepness such that they never cross, we call these parallel. When two lines meet at a 90 degree angle, it is called perpendicular. Let’s suppose that line “a” has a slope of 72 ; it is pretty steep and positive. Line “b” will have a similar ratio, but we can see that it is shallow and negative. As we can see from the picture, the slope of “b” is m= - 72 . This pattern happens every time that two lines are perpendicular. As a rule: A perpendicular slope is the negative reciprocal. Some people like to think of it that the
a
7 -2 7
2
b
Section 4.3
167
two slopes will always multiply together to give you -1. As a special case, can you see what slope would be perpendicular to 0? Since vertical and horizontal are perpendicular, an undefined slope is the answer.
Here are a few examples: Equation
Slope
Parallel slope
Perpendicular
3x+2y=7
m= - 32
m= - 32
m= 23
y = 5x-2
m=5
m=5
m= - 15
4x-7y=7
m= 47
m= 74
m= - 74
x=-7
m = undefined
m = undefined
m=0
y=3
m=0
m=0
m = undefined
168
Section 4.3 Exercises Part A 1. Three types of horses are in a local ranch. The number of Arabians is 8 more than twice the
number of Quarter-horses, and the number of Clydesdales is50 more than the number of Quarterhorses. There are a total of 282 horses at the ranch. How many of each kind are there? 2
2. What is the slant height of a cone that has Surface Area of 219.91 in and a radius of 5 in? 3. The perimeter of a rectangle is 120 in. If the length of the rectangle is 3 more than twice the
width, what are the dimensions of the rectangle? 4.
Original Price:$392.50 Tax: 6% Final Price:
5.
Original Price: Tax: 7% Final Price: $73.90
Fill out the table for each of the following: 6. 2x - 3y = 9
x 5 -4
y
3 0 7
7.
y = x+2
x 2 0 -1
y
0 4
Graph the following lines, and label x and y intercepts.
8.
5x + 2y = 10
11. x = 10
9. y = x - 6
10. y = x
12. y = - 37 x +4
13. 7x – y = 14
Find the slope between each pair of points. 14. (8,-2) (7,3)
15. (8,1) (-5,6)
16. (-3,-1) (-3,-8)
17. (7,9) (-2,3)
18. (-5,2) (4,6)
19. (-6,1) (6,1)
Assignment 4.3a
169
Graph the following lines giving one point and the slope.
Ex. 2x – 7y = 3
Find one point: ( 32 ,0) and the slope: m = 72 . Then graph the point. Then go up 2 and over 7 for the next one:
7 2
.
21. y = 4x + 3
22. y = x - 4
23. x = -6
24. y = - 37 x - 2
25. 3x – 4y = 12
26.
27. x + 4y = 9
28. y = 7
20.
-6x + y = 10
5x + 3y = 10
Preparation 29. Write down 5 equations of lines that have the slope:
m= -
Assignment 4.3a
170
Answers: 56 Quarter-horses, 106 1. Clydesdales, 120 Arabian slant height = 9 in 2.
3.
41in X 19in
4.
$416.05
5.
$69.07
6.
x 5 -4 9
15
7.
x 2 0 -1 -
10.
(0,0) (3,8)
20.
(0,10); m = 6
11.
(10,0)
21.
(0,3); m = 4
12.
(0,4) ( ,0)
22.
(0,-4); m =
no y-int
y
-
3 0 7 y 9 2 - 0 4
8.
(0,5) (2,0)
13.
(2,0) (0,-14)
23.
(-6,0); m = undefined
9.
(0,-6) ( ,0)
14. 15. 16. 17. 18.
m = -5 m = - m = undefined m = m = m=0
24.
(0,-2); m = -
19.
Assignment 4.3a
171
25.
(4,0); m =
27. (9,0); m = -
26.
(2,0); m = -
28. (15,7); m = 0
29. Discuss it together.
Assignment 4.3a
172
Slope Monster Equation 2x – 5y = 7
Slope
Equation 4x – y = 7
y= x-4
y= x-4
5x – 3y = 7
8x – 3y = 12
2x + 7y = 19
- 4x + 7y = 19
x = 13
x = -19
y= x-8
y= x-4
y = 5x – 8
y = -3x – 8
-3x + 9y = 4
-10x + 6y = 4
y = -3
y = 15
y=-
x-4
7x – 3y = 7
y=
x-4
2x – 8y = 17
y= x-4
y= x+6
5x – 3y = 7
4x + 7y = 7
4x + 7y = 19
2x - 9y = 19
x=-3
x=7
y=- x-4
y= x-4
y = -2x – 8
y = 4x + 13
-3x + 6y = 4
-3x - 6y = 4
y = -5
y=7
y=- x-4
Slope
y = - x + 15
Assignment 4.3 Slope Monster
173
Slope Monster Solution Equation 2x – 5y = 7
y= x-4
Slope m= m=
Equation
4x – y = 7
m=4
y= x-4
m=
8x – 3y = 12
m=
Slope
5x – 3y = 7
m=
2x + 7y = 19
m = −
- 4x + 7y = 19
m =
x = 13
Undefined
x = -19
Undefined
y= x-8
m=
y = 5x – 8
m=5
m=
y = -3
m=0
x-4
7x – 3y = 7
y= x-4
5x – 3y = 7
m=− m= m= m=
x=-3
Undefined
y=- x–4
m=−
y = -2x – 8
m=-2 m=
y = -5
m=0
y=- x-4
-3x + 6y = 4
m=−
m = -3
-10x + 6y = 4
m=
y = 15
m=0
y=
x-4
m= m=
y= x+6
m=
4x + 7y = 7
m=−
m=−
y = -3x – 8
4x + 7y = 19
m=
2x – 8y = 17
y= x-4
-3x + 9y = 4
y=-
2x - 9y = 19
m=
x=7
Undefined
y= x-4
m=
y = 4x + 13
m =4
-3x - 6y = 4
m=−
y=7
m =0
y = - x + 15
m=−
174
Section 4.3 Exercises Part B
Fill out the table for each of the following: 1. 2x - 5y = 11
x 5 -4
2.
y = x+6
y
3 0 7
x 2 0 -1
y
0 4
Graph the following lines, and label x and y intercepts.
3.
4. y = - x - 6
4x - 2y = 10
5. y = 5x
Find the slope between each pair of points. 6. (3,-2) (7,3)
7. (9,1) (-7,6)
8. (5,-1) (-3,-8)
9. (-2,9) (-2,3)
10. (-5,2) (5,6)
11. (19,1) (6,1)
12. Explain the difference between a slope of zero and an undefined slope. Graph the following lines giving one point and the slope. 13.
-3x + 4y = 10
16. y = 17
14. y = 2x - 7
15. y = x - 4
17. y = - 37 x - 2
18. 2x – 6y = 12
Write the equations of the lines with the slopes and points: Ex. Write an equation of the line that has slope m = answer in Standard Form.
From the slope m =
4 7
4 7
, and goes through the point (2,1). Put the
, I know that the equation must look like: 4x – 7y = something, so I put in the point to see what it is. 4(2) – 7(1) = 1.
Assignment 4.3b
175
Thus the answer is 4x – 7y = 1. Ex. Write an equation of the line that has slope m =
4 7
, and goes through the point (2,1). Put the
answer in Slope-Intercept Form.
From the slope m =
4 7
, I know that the equation must look like: y=
4 7
x+b
1=
4 7
(2) + b
Put the point in to see what b is.
1- 87 = b - 17 = b Thus the answer is y =
4 7
x-
1 7
.
19. Write an equation of the line that has slope m= -3, and goes through the point (-4,6). Put the
answer in Standard Form.
20. Write an equation of the line that has slope m= 58 , and goes through the point (3,6). Put the
answer in Standard Form.
21. Write an equation of the line that has slope m=- 23 , and goes through the point (1,-3). Put the
answer in Slope-Intercept Form.
22. Write an equation of the line that has slope m=- 54 , and goes through the point (5,-3). Put the
answer in Slope-Intercept Form.
23. Write an equation of the line that has slope m= 2, and goes through the point (0,5). Put the
answer in Slope-Intercept Form.
24. Write an equation of the line that has slope m=- 17 , and goes through the point (-4,7). Put the
answer in Standard Form.
Assignment 4.3b
176
Answers: 1.
6.
y
13
3 0
10.
7
12.
Undefined is vertical 0 is horizontal
13.
(0,) m =
23
2.
x 2 0 -1 -
-
3.
m = m = - m = m = undefined m = m=0
x 5 -4
-
7.
8.
9. 11.
y 13 6
18.
(6,0) m =
19.
3x + y = -6
20. 21.
5x – 8y = -33 y = - x - y = - x + 1 y = 2x + 5 x + 7y = 45
22. 23.
24.
0 4
14.
(0,-7) m = 2
15.
(0,-4) m =
(0,-5) (,0)
4.
5.
(0,-6) (- ,0)
16.
(0,17) m = 0
17.
(0,-2) m = -
(0,0) (2,10)
Assignment 4.3b
177
Section 4.3 Exercises Part C – Exam Review 1. Chart the following table of growth of a savings account:
# of 0 years Amount 35
1
2
3
4
5
6
7
8
50
75
102
130
161
174
205
240
Based on the chart:
2. In what year did the
company make $450? 3. How much did the
company make in 2006? 4. What years did the
company make over $500?
5. Graph the growth of a savings account in Excel, using the savings formula, over the course of
20 years of a savings account that starts out at $200 and adds $50 per month and gets 7% interest.
Fill out the table for each of the following: 6. 3x + 4y = 11
x 5 -4
y
3 0 7
y = x - 2
7.
x 2 0 -1
y
0 4
Graph the following lines, and label x and y intercepts.
8.
5x - y = 10
9. y = - x - 5
10. y = -2x
Assignment 4.3c
178
Find the slope between each pair of points. 11. (4,-2) (7,3)
12. (3,1) (-7,6)
13. (5,-1) (5,-8)
14. (-2,9) (-2,254)
15. (-5,2) (5,7)
16. (19,1) (6,2)
17. Explain the difference (again) between a slope of zero and an undefined slope. Graph the following lines giving one point and the slope.
18.
-3x + 5y = 10
21. y = -5
19. y = - x - 2
20. y = x - 1
22. y = - x - 2
23. 12x – 6y = 12
Write the equations of the lines with the slopes and points: 24. Write an equation of the line that has slope m = 2, and goes through the point (-4,1). Put the
answer in Standard Form.
25. Write an equation of the line that has slope m =
,and goes through the point (-14,6). Put the
answer in Standard Form.
2
26. Write an equation of the line that has slope m = - 3 , and goes through the point (0,0). Put the
answer in Slope-Intercept Form.
27. Write an equation of the line that has slope m = -
, and goes through the point (2,-3). Put
the answer in Slope-Intercept Form.
28. Write an equation of the line that has slope m = 7, and goes through the point (0,5). Put the
answer in Slope-Intercept Form.
29. Write an equation of the line that has slope m = - , and goes through the point (-4,7). Put
the answer in Standard Form. 30. Create a Visual Chart on one side of a piece of paper for Chapter 4 material including information and examples relating to Charts, Graphs, and Lines.
Assignment 4.3c
179
Answers: 1. 9.
10. 2. 3. 4. 5.
x 5 -4 -
0 7
x 2
y
0 -1
-2
8.
22.
(0,-2) m = -
23.
(1,0) m = 2
24. 25. 26. 27. 28. 29.
2x – y = -9 3x – 7y = -84 y = - x y = - x + 2 y = 7x + 5 x + 4y = 24
30.
Make it nice.
(0,0) (2,-4)
3
7.
(0,-5) m = 0
21.
y -1
-
(0,-1) m =
(0,-5) (- ,0)
2003 about $340 2004(?), 2009, 2010 On Spreadsheet
6.
20.
- 0
11. 12. 13. 14. 15. 16.
m = m = - m = undefined m = undefined m = m=-
17.
Undefined is vertical 0 is horizontal
18.
(0,2) m =
4
(0,-10) (2,0) 19.
(0,-2) m = -1
Assignment 4.3c
