Let Math April

LICENSURE EXAMINATION for TEACHERS 2012 Refresher Course - GENERAL EDUCATION MATHEMATICS (Lecture)

WHAT TO EXPECT GENERAL EDUCATION FOCUS: MATHEMATICS Use of four fundamental Operations in problem solving 1. Operations with whole numbers, decimals, fractions and integers 2. Least common multiple; greatest common factor 3. Divisibility rules 4. Ratio and proportion 5. Percentage, rate and base

6. Measurement and units of measure a. Perimeter b. Area c. Volume d. Capacity e. Weight 7. Number Theory a. Prime and composite numbers b. Prime factorization

Prepared by: MR. VON ANTHONY G. TORIO

GENERAL TIPS FOR SOLVING MATHEMATICS ITEMS 1. Draw a diagram figure and subtract from it the area of the unshaded region.  For any geometry question for which a figure is not provided, draw one (as accurately as possible) – never attempt a geometry problem 6. Don’t do more than you have to without first drawing a diagram.  Very often a problem can be solved in more than one way. You should always try to do it in the 2. Trust a diagram that has been drawn to scale easiest way possible.  Whenever diagrams have been drawn to scale, they can be trusted. This means that you can 7. Pay attention to units look at the diagram and use your eyes to  There are instances where items given are not accurately estimate the sizes of angles and line consistent with the units in the options. There segments. will be a need for you to change the unit of your answer. It will be good to watch over units and 3. Exaggerate or otherwise change a diagram be consistent in using them.  Sometimes it is appropriate to take a diagram that appears to be drawn to scale and 8. Systematically make lists intentionally exaggerate it. Why would we do  When a question asks “how many” often the best this? strategy is to make a list of all the possibilities. If 4. Add a line to a diagram you do this it is important that you make the list in a systematic fashion so that you don’t  Occasionally, after staring at a diagram, you still inadvertently leave something out. Usually, this have no idea how to solve the problem to which means listing the possibilities in numerical or it applies. It looks as though not enough alphabetical order. Often, shortly after starting information has been given. When this happens, the list, you can see a pattern developing and it often helps to draw another line in the you can figure out how many more entries there diagram. will be without writing them all down. Even if the question does not specifically ask “how 5. Subtract to find the shaded region many” you may need to count something to  Whenever part of a figure is shaded, the answer it: in case, as well, the best plan may be straightforward way to find the area of the to write out a list. shaded portion is to find the area of the entire LET Review 2011 Focus: Mathematics

Page 1 Prepared by Mr. Von Anthony G. Torio

ADDITIONAL TIPS  Round off and estimate wherever possible. Minimize calculation.  It is easier to divide and use cancellation method than to multiply first and later divide. This will help make quick calculations.  Look for shortcuts that are built into many problems.  Work on units (hours, ounces, square miles) consistent with those in the answer choices.  Break down complex word problems and make computations one step at a time.  Consider using labels and symbols to represent long sentences, statements, and ideas. This will help you have a good grasp of the problem.

   





Consider what is asked and focus on it, do not spend time completing your solution. Since most tests are in multiple choice formats, consider the options. Go through the entire test, doing only the problems that can be solved easily. Go back through the test, doing the problems that seem to be on familiar material but will take a little time to figure out. Finally, if there is time, use answer choices as a guide in re-examining items previously answered. At each step, be careful to mark the correct space on the answer sheet.

Number System Sets of Numbers (1) Even Numbers = {2, 4, 6,..} (2) Odd Numbers = {1, 3, 5, 7,…} (3) Positive Integers = {1,2,3,…} (4) Negative Integers = {…, -3, -2, -1} (5) Whole Numbers (W) = {0, 1, 2, 3,…} (6) Integers (Z) = {…, -3, -2, -1, 0, 1, 2,3,…} (7) Rational Numbers (Q) = {x/x = ; a, b Z, b 0} Properties of Real Numbers Least Common Multiple (LCM) – the smallest natural number that has both of them as factor. Example: 12 is the LCM of 3 and 4. Least Common Denominator (LCD) – the smallest integer that contains both the denominators as factor. Fractions/Decimals/Percentages Fractions are numbers expressed as the ratio of two numbers, and are used primarily to express a comparison between parts and a whole. A kind of fraction, percentage, always has denominator is always 100. Thus 75% means 75/100. In mathematics, the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero, is called the set of rational numbers. This set is represented by the symbol Q. Vulgar, proper, and improper fractions A vulgar fraction (or common fraction or simple fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator) such as . A vulgar fraction is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator—that is, if the absolute value of the entire fraction is less than 1; a vulgar fraction is said to be an improper fraction if the absolute value of the numerator is

(8) Prime Numbers = {2, 3, 5, 7, 11, 13, 17, 19,…} (9) Real Numbers (R)= set of all rational and irrational numbers (10) Natural Numbers (N) – a.k.a. Counting Numbers = {1,2,3,…} (11) Irrational Numbers = Set of numbers which cannot be expressed in the form ; where a and b

Z, b 0}

Example: ¾ and 5/7, the LCD is 28. Since 28 has for its factors, 7 and 4. Greatest Common Factor (GCF) – refers to the largest integer which is a factor of each of the given number. Example: the factors of 40 (5,23) and 70(7,2 and 5). The common factors are 2 and 5. The GCF is 5.

greater than or equal to the absolute value of the denominator (e.g. ) Mixed numbers A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+"; the whole and fractional parts of the number are written next to each other: . Reciprocals and the "invisible denominator" The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of , for instance, is . Because any number divided by one results in the same number, it is possible to write any whole number as a fraction by using 1 as the denominator: 17 = (1 is sometimes referred to as the "invisible denominator"). Therefore, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be

LET Review 2012

Focus: Mathematics

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Prepared by Mr. Von Anthony G. Torio

Place Value

Conversion factors

Metric Prefixes

Additional: 1 score = 20 years ; 1 mL = 1 cm3

Polynomials Polynomial (from Greek poly, "many" and medieval Latin binomium, "binomial”) is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not a whole number (3/2).

Exponents and Radicals Laws of Exponent a0 = 1 aman = am+n

n

=

= =√ = amam

(ab)m

LET Review 2012

Focus: Mathematics

m=

Laws of Radicals √

= √

√ =

√ √

√√ = √

√

A polynomial is a sum of terms. For example, the following is a polynomial:

the degree of a non-zero polynomial is the largest degree of any one term

Factors and Products a(x + y) = ax + ay (x + y)2 = x2 + 2xy + y2 (x – y)2 = x2 – 2xy + y2 (x – y) (x + y) = x2 – y2 (x3 + y3) = (x + y) (x2 – xy + y2) (x3 – y3) = (x – y) (x2 + xy + y2) (x + y)3 = x3 + 3xy2 + 3xy2 + y3 (x – y)3 = x3 – 3x2y + 3xy2 – y3 Page 3


Solving Equations in One variable Example 1 2x + 7 = 3x 7 = 3x – 2x ; combining like terms 7=x ; after simplifying

Example 3 2x + 7 = 3 – 3x 2x + 3x = 3 + 7

; combining like terms, transposing terms with

x to one side and those without to the other

5x = 10 ; after simplifying Example 2 (5x / 5) = 10/5; dividing both sides with the coefficient of x 4y – 7 = 1 4y = 1 + 7 ; combining like terms, transposing 7 to the x = 2 ; after simplifying other side 4y = 8 ; after simplifying, (adding 1 and 7) 4y / 4 = 8 / 4 ; dividing both sides with the coefficient of y, to leave y y = 2 ; after simplifying LINEAR EQUATIONS WITH TWO VARIABLES Options: 1. Elimination 2. Substitution USING ELIMINATION METHOD

3x - 3y = 3 7x =7 (7x/7) = (7/7) x=1 * through inspection, the two equations may be added to solve for x.

Example 1 2x + 3y = 4 ; equation 1 2x + 2y = 1 ; equation 2

After solving for x, either of the two equations may be solved to solve for y; using equation 2 solve for y: 3x - 3y = 3 3(1) – 3y = 3 Using the elimination method: 3 – 3y = 3 2x + 3y = 4 3y = 3 – 3 2x + 2y = 1 3y = 0 y=3 (3y / 3) = (0 / 3) * through inspection, the two equations may be y=0 subtracted to solve for y. Final Answers: x = 1 ; y = 0 After solving for y, x could be solved using either of the two equations. Considering the first equation: 2x + 3y = USING SUBSTITUTION METHOD 4 and substituting the value of y: 2x + 3y = 4 Example 1 2x + 3(3) = 4 4x + y = 1 ; equation 1 2x + 9 = 4 3x + y = 3 ; equation 2 2x = 4 – 9 2x = 5 Substitution means that an equation will be used to (2x/2) = (5/2) replace one-variable to the other equation. This leaves x = 5/2 one-variable for the combination of the equations. This makes the equation as an equation in one-variable. Final Answers: x = 5/2 ; y = 3  By inspection, y can be used to for substitution Example 2 4x + 3y = 4 ; equation 1 4x + y = 1 y = 1 – 4x ; this new form of the 3x - 3y = 3 ; equation 2 equation can now be used for substitution to the second equation: Using the elimination method: 3x + y = 3 3x + (1 – 4x) = 3 4x + 3y = 4 3x – 4x = 3 - 1 ; combining like terms LET Review 2012

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(-x)(-1) = (2)(-1) x=-2 Solving for y

-x = 2 ; simplifying ;multiplying both sides by -1 ; after simplifying

*using this equation for substitution to the second equation

2x + 2y = 3 2 (2 – 4y) + 2y = 3 4 – 4y + 2y = 3 ; after distribution Since one of the variables is already known, either of 4 – 4y = 3 ; after simplifying the initial two equations may be used to solve for the - 4y = 3 – 4 ; combining like terms other variable: using equation 2: 3x + y = 3 and - 4y = -1 ; simplifying substituting the value of x: (-4y)/(-4) = (-1)/(-4) ; dividing both sides by (-4) y=¼ 3(-2) + y = 3 Since one of the variables is already known, either of -6 + y = 3 the initial two equations may be used to solve for the y=3+6 other variable: y=9 using equation (1): x + 4y = 2 Final Answers: x = -2 ; y = 9 x + 4(1/4) = 2 x+1=2 ; after simplifying x=2–1 ; combining like terms Example 2 x=1 ; after simplifying x + 4y = 2 ; equation 1 2x + 2y = 3 ; equation 2 Final Answers: x = 1; y = 1/4 * through inspection; the first equation may be used for substitution to the second equation: x + 4y = 2 ; equation 1

x = 2 – 4y

CLOCK PROBLEM Clock angle problems relate two different measurements: angles and time. To answer the problem the relationship between the time shown (or an elapsed time) and the position of the hands (as given by an angle) has to be found. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute.

Equation for the degrees on the minute hand θmin. = 6M where:  

is the angle in degrees of the hand measured clockwise from the 12 o'clock position. is the minute.

Example The time is 5:24. The angle in degrees of the hour hand is:

The angle in degrees of the minute hand is:

Equation for the angle of the hour hand Equation for the angle between the hands where:    

The angle between the hands can also be found using the formula:

is the angle in degrees of the hand measured clockwise from the 12 o'clock position. is the hour. is the minutes past the hour. is the minutes past 12 o'clock.

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Example The time is 2:40.

Hour and minute hands are superimposed only when their angle is the same.

Where:  is the hour  is the minute When are hour and minute superimposed?

hands of a clock

is an integer in the range 0–11. This gives times of: 0:00, 1:05.45, 2:10.90, 3:16.36, etc.

Interest problems Interest = Principal x interest rate per period x period I = Prt

Where: I = interest , P – principal amount, r – rate, t – time

Note: the unit for rate should be consistent with time. Example if r = 5%/year; time should be in years Work Problems For the case where two persons/instruments/machines are working to achieve a single goal ;

Where: t – total time

x and y – person’s/instrument’s/machines’ time in working for a goal For the case where two persons/instruments/machines are working against each other. The goal of the other is the reverse of another. (e.g., one pipe fills a pool and the other pipe drains the pool) ;

Where: t – total time

x and y – person’s/instrument’s/machines’ time in working for a goal Motion Problem Speed = distance/time s = d/t where: s – speed, d – distance and t - time IMPORTANT PLANE GEOMETRY FORMULAS FIGURE FORMULA/concepts FIGURE FORMULA/concepts Triangle Circle Perimeter (P) – sum of Area (A): A = πr2 sides (s): P = s1 + s2 + s3 Circumference (C): C = 2 πr = πD Area (A): A = (b)(h) Isosceles triangle: triangle with two sides Circle and two angles equal

Triangle a

Right triangle: a triangle with one of the angles equal to 90 degrees.

c

b Parallelogram

LET Review 2012

Focus: Mathematics

Perimeter (P) = 4s

Follows Pythagorean Square theorem: c2 = a2 + b2

b Right Triangle a

Where: r - radius and D – diameter Square Area (A): A = s2

h

Parallelogram Area (A) = bh Perimeter (P) = 2a + 2b Where: a & b – sides h - height

b2 a2

a1 b1 Trapezoid

Trapezoid Perimeter (P): P = a1 + a2 + b 1 + b 2 Area (A): A=

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IMPORTANT SOLID GEOMETRY FORMULAS FIGURE FORMULA/concepts Rectangular Prism Rectangular Prism Total Surface Area (TSA): TSA = 2lw + 2wh + 2lh h Volume (V): l w V = lwh Right Circular Cylinder

h Circular base Right Circular Cone s r Circular base

FIGURE Cube

e

Where: e - edge

Total Surface Area (TSA): TSA = 2πr2 + 2 πrh Volume (V): V = πr2h

Rectangular Prism

Right Circular Cone

Rectangular base

FORMULA/concepts Cube Total Surface Area (TSA): TSA = 6e2 Volume = e3 = s3

Rectangular Prism Volume (V): V = Abaseh Where: Abase – Area of the base h - height

Total Surface Area (TSA): TSA = πr2 + πrs Volume (V): V = (πr2)(h)

Probability and Statistics Range = HS – LS + 1 Where: HS – Highest Score LS – Lowest Score Arithmetic mean (AM)

Harmonic mean (HM)

The harmonic mean is an average which is useful for The arithmetic mean is the "standard" average, often simply sets of numbers which are defined in relation to some unit, called the "mean". for example speed (distance per unit of time).

For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is

For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is

Geometric mean (GM) The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic Measures of Central Tendency mean) e.g. rates of growth. Arithmetic Mean The arithmetic mean is the most common measure of central tendency. It simply the sum of the numbers divided by the number of numbers. The symbol mm is used for the mean of a population. The symbol MM is used for the mean of a For example, the geometric mean of six values: 34, 27, sample. The formula for mm is shown below: m=Σ⁢XN m Σ X 45, 55, 22, 34 is: N where Σ⁢X Σ X is the sum of all the numbers in the numbers in the sample and NN is the number of numbers in the sample. As an example, the mean of the numbers 1+2+3+6+8=205=4 1 2 3 6 8 20 5 4 regardless of whether LET Review 2012

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the numbers constitute the entire population or just a Computation of the Median: When there is an odd number of sample from the population. numbers, the median is simply the middle number. For example, the median of 2, 4, and 7 is 4. When there is an even The table, Number of touchdown passes, shows the number number of numbers, the median is the mean of the two of touchdown (TD) passes thrown by each of the 31 teams in middle numbers. Thus, the median of the numbers 22, 44, 77, the National Football League in the 2000 season. The mean 1212 is 4+72=5.5 4 7 2 5.5 . number of touchdown passes thrown is 20.4516 as shown below. m=Σ⁢XN=63431=20.4516 m Σ X N 634 31 20.4516 Mode 37

33

33

32

29

28

28

23

The mode is the most frequently occuring value. For the data in the table, Number of touchdown passes, the mode is 18 22 22 22 21 21 21 20 20 since more teams (4) had 18 touchdown passes than any other number of touchdown passes. With continuous data 19 19 18 18 18 18 16 15 such as response time measured to many decimals, the frequency of each value is one since no two scores will be 14 14 14 12 12 9 6 exactly the same (see discussion of continuous variables). Therefore the mode of continuous data is normally Table 1: Number of touchdown computed from a grouped frequency distribution. The Grouped frequency distribution table shows a grouped passes frequency distribution for the target response time data. Since the interval with the highest frequency is 600-700, the Although the arithmetic mean is not the only "mean" mode is the middle of that interval (650). (there is also a geometic mean), it is by far the most commonly used. Therefore, if the term "mean" is used Range Frequency without specifying whether it is the arithmetic mean, the geometic mean, or some other mean, it is assumed to refer to 500-600 3 the arithmetic mean. 600-700 6 Median 700-800 5 The median is also a frequently used measure of central tendency. The median is the midpoint of a distribution: the same number of scores are above the median as below it. For the data in the table, Number of touchdown passes, there are 31 scores. The 16th highest score (which equals 20) is the median because there are 15 scores below the 16th score and 15 scores above the 16th score. The median can also be thought of as the 50th percentile.

800-900

5

900-1000

0

1000-1100

1

Table 3: Grouped frequency distribution

Let's return to the made up example of the quiz on which you made a three discussed previously in the module Permutation and Combination Introduction to Central Tendency and shown in Table 2. Permutations Student

Dataset 1

Dataset 2

Dataset 3

You

3

3

3

John's

3

4

2

Maria's

3

4

2

Shareecia's

3

4

2

Luther's

3

5

1

Table 2: Three possible datasets for the 5-point make-up quiz For Dataset 1, the median is three, the same as your score. For Dataset 2, the median is 4. Therefore, your score is below the median. This means you are in the lower half of the class. Finally for Dataset 3, the median is 2. For this dataset, your score is above the median and therefore in the upper half of the distribution. LET Review 2012

Focus: Mathematics

BY THE PERMUTATIONS of the letters abc we mean all of their possible arrangements: abc acb bac

bca cab cba

There are 6 permutations of three different things. As the number of things (letters) increases, their permutations grow astronomically. For example, if twelve different things are permuted, then the number of their permutations is 479,001,600. Now, this enormous number was not found by counting them. It is derived theoretically from the Fundamental Principle of Counting: If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened,

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something else can be chosen in n different ways, then the number of ways of choosing both of them is m · n. For example, imagine putting the letters a, b, c, d into a hat, and then drawing two of them in succession. We can draw the first in 4 different ways: either a or b or c or d. After that has happened, there are 3 ways to choose the second. That is, to each of those 4 ways there correspond 3. Therefore, there are 4· 3 or 12 possible ways to choose two letters from four. ab ba ca da ac

bc

cb

db

ad

bd

cd

dc

eq. Therefore, the total number of ways they can be next to each other is 2· 5! = 240. Permutations of less than all We have seen that the number of ways of choosing 2 letters from 4 is 4· 3 = 12. We call this "The number of permutations of 4 different things taken 2 at a time." We will symbolize this as 4P2: 4P 2

The lower index 2 indicates the number of factors. The upper index 4 indicates the first factor. For example, 8P3 means "the number of permutations of 8 different things taken 3 at a time." And

ab means that a was chosen first and b second; ba means that b was chosen first and a second; and so on.

8P3

Let us now consider the total number of permutations of all four letters. There are 4 ways to choose the first. 3 ways remain to choose the second, 2 ways to choose the third, and 1 way to choose the last. Therefore the number of permutations of 4 different things is 4· 3· 2· 1 = 24 Thus the number of permutations of 4 different things taken 4 at a time is 4!. (To say taken 4 at a time is a convention. We mean, "4! is the number of permutations of 4 different things taken from a total of 4 different things.") In general, The number of permutations of n different things taken n at a time is n!. Example 1. Five different books are on a shelf. In

= 8· 7· 6 = 56· 6 = 50· 6 + 6· 6 = 336

For, there are 8 ways to choose the first, 7 ways to choose the second, and 6 ways to choose the third. In general, nPk

= n(n − 1)(n − 2)· · · to k factors Factorial representation

We saw in the Topic on factorials, 8! = 8· 7· 6 5! 5! is a factor of 8!, and therefore the 5!'s cancel. Now, 8· 7· 6 is 8P3. We see, then, that 8P3 can be expressed in terms of factorials as

how many different ways could you arrange them?

8P3

=

Answer. 5! = 1· 2· 3· 4· 5 = 120 Example 2. There are 6! permutations of the 6 letters of the word square. a) In how many of them is r the second letter? _ r _ _ __ b) In how many of them are q and e next to each other? Solution. a) Let r be the second letter. Then there are 5 ways to fill the first spot. After that has happened, there are 4 ways to fill the third, 3 to fill the fourth, and so on. There are 5! such permutations. b) Let q and e be next to each other as qe. Then we will be permuting the 5 units qe, s, u a, r.. They have 5! permutations. But q and e could be together as LET Review 2012

Focus: Mathematics

= 4· 3

8! 8! = (8 − 3)! 5!

In general, the number of arrangements -permutations -- of n things taken k at a time, can be represented as follows: nPk

=

n! . . . . . . . . . . . .(1) (n − k)!

The upper factorial is the upper index of P, while the lower factorial is the difference of the indices. Example 3. Express Solution.

10P4

=

10P4

in terms of factorials.

10! 6!

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The upper factorial is the upper index, and the lower factorial is the difference of the indices. When the 6!'s cancel, the numerator becomes 10· 9· 8· 7. This is the number of permutations of 10 different things taken 4 at a time. Example 4. Calculate nPn. Solution.

nPn

=

n! n! n! = = = n! (n − n)! 0! 1

nP n

is the number of permutations of n different things taken n at a time -- it is the total number of permutations of n things: n!. The definition 0! = 1 makes line (1) above valid for all values of k: k = 0, 1, 2, . . . , n.

many permutations as combinations. 4C3 , therefore, will be 4P3 divided by 3! -- the number of permutations that each combination generates. 4· 3· 2 4C3 = = 1· 2· 3 3! Notice: The numerator and denominator have the same number of factors, 3, which is indicated by the lower index. The numerator has 3 factors starting with the upper index and going down, while the denominator is 3!. In general, nCk =

nC

k

=

.

k!

n(n − 1)(n − 2)· · · to k factors k!

Example 1. How many combinations are there of 5 In permutations, the order is all important -- we count abc as different from bca. But in combinations we are concerned only that a, b, and c have been selected. abc and bca are the same combination. Here are all the combinations of abcd taken three at a time: abc abd acd bcd. There are four such combinations. We call this The number of combinations of 4 distinct things taken 3 at a time. We will denote this number as 4C3. In general, nCk

= The number of combinations of n distinct things taken k at a time.

Now, how are the number of combinations nCk related to the number of permutations, nPk ? To be specific, how are the combinations 4C3 related to the permutations 4P3? Since the order does not matter in combinations, there are clearly fewer combinations than permutations. The combinations are contained among the permutations -they are a "subset" of the permutations. Each of those four combinations, in fact, will give rise to 3! permutations: abc abd acd bcd acb

adb

adc

bdc

bac

bad

cad

cbd

bca

bda

cda

cdb

cab

dab

dac

dbc

cba

dba

dca

dcb

Each column is the 3! permutations of that combination. But they are all one combination -- because the order does not matter. Hence there are 3! times as LET Review 2012

Focus: Mathematics

distinct things taken 4 at a time? Solution.

5C4

=

5· 4· 3· 2 = 5. 1· 2· 3· 4

Again, both the numerator and denominator have the number of factors indicated by the lower index, which in this case is 4. The numerator has four factors beginning with the upper index 5 and going backwards. The denominator is 4!. Example 2. Evaluate 8C6. Solution.

8C6

=

8· 7· 6· 5· 4· 3 = 28. 1· 2· 3· 4· 5· 6

Both the numerator and denominator have 6 factors. The entire denominator cancels into the numerator. This will always be the case. Example 3. Evaluate 8C2. Solution.

8C2

=

8· 7 = 28. 1· 2

We see that 8C2 , the number of ways of taking 2 things from 8, is equal to 8C6 (Example 2), the number of ways of taking 8 minus 2, or 6. For, the number of ways of taking 2, is the same as the number of ways of leaving 6 behind. Always: nCk

= nCn − k

The bottom indices, k on the left and n − k on the right, together add up to n. Example 4. Write out nC3.

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Solution. nC 3 =

n(n − 1)(n − 2) 1· 2· 3

The 3 factors in the numerator begin with n and go down.

Note also the convention that the factorial of the lower index, k, is written in the denominator on the right.

Example 5. 8C3 =

8! . (Note: 5 + 3 in the denominator 5! equals 8 in 3! the numerator.

Factorial representation In terms of factorials, the number of selections -combinations -- of n distinct things taken k at a time, can be represented as follows: n! nC = k (n − k)! k! This is nPk divided by k!. Compare line (1) of Section 1.

Show that this is equal to

8· 7· 6 . 1· 2· 3

Solution. 5! is a factor of 8!, so it will cancel. 8! 8· 7· 6· 5! 8· 7· 6 = = . 5! 3! 5!· 1· 2· 3 1· 2· 3 Final Answer: 56

Notice: In the denominator, n − k and k together equal the numerator n.

Additional Concepts Calendar Dates:  Months with 30 days: September, April, June and November  Months with 31 days: January, March, May, July, August, October, December  February has either 28/29 days. The 29th day is added during leap years. Every four years. Years that are divisible by 4.

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LICENSURE EXAMINATION for TEACHERS 2012 Refresher Course - GENERAL EDUCATION MATHEMATICS (Previous LET Items)

WHAT TO EXPECT GENERAL EDUCATION FOCUS: MATHEMATICS Use of four fundamental Operations in problem solving 1. Operations with whole numbers, decimals, fractions and integers 2. Least common multiple; greatest common factor 3. Divisibility rules 4. Ratio and proportion 5. Percentage, rate and base

6. Measurement and units of measure a. Perimeter b. Area c. Volume d. Capacity e. Weight 7. Number Theory a. Prime and composite numbers b. Prime factorization

Prepared by: MR. VON ANTHONY G. TORIO LET SEPTEMBER 2010 items

8. Only 682 examinees passed in a Nurse Licensure Examination out of 2550 examinees. What is the passing 1. Two times a certain number added to 40 gives the same percentage? result as three times the same number subtracted from 75. a. 26% b. 26.7% c. 26.8% d. 26.5% What is the number? a. 7 b. 30 c. 15 d. 10 9. To find perimeter (P) of a rectangle that has length of 16 inches and width of 12 inches which of the following 2. A pancake was cut into 8 pieces. Three brothers ate one equation can be used? piece each. What part of the cake was left? a. P = 2(16)(12) c. P = 2(16) + (12) a. ½ b. 2/8 c. 5/8 d. ¼ b. P = (16) + (12) d. P = 2(16+12) 3. A meat dealer sold his butchered cow for P 21, 500. If each 10. A swimming pool is an equilateral triangle in shape. One kilogram of beef cost P120 each, how many kilograms side is 11 meters. How many meters rope are needed to were realized from one cow? enclose the pool? a. 179.16 kg c. 180 kg a. 55 meters c. 44 meters b. 179.18 kg d. 178 kg b. 45 meters d. 33 meters 4. How many twenty thousand are there in one million? a. 100 b. 500 c. 50 d. 1000 5. What is the cube root of (32 x 3)? a. 9 b. 3 x 3 c. 27

d. 3

11. Mrs. Alice’s rectangular bathroom has to be covered with tiles. The edge of the bathroom needs a rubberized tile. If the tub is 2.3 meters long and 1.8 meters wide, how many meters of rubberized tiles are required? a. 9.2 meters c. 8.2 meters b. 4.6 meters d. 2.3 meters

6. Which expresses the polynomial in the following? a. 2x + 4x – 3 c. x – x + 9 – 3x + 10x b. xy – x – 6 d. x – 4x + 2x – x + 2

12. An executive office has to be carpeted. The area is 3 m by 4 m. The carpet costs P 1, 000.00 per square meter. How much will be spent for the purchase of the carpet? 7. A swimming contest is held in a four lane swimming pool a. P120, 000.00 c. P10, 000.00 of 50 meters long. The contest is for the 200 meter. How b. P12, 000.00 d. P 1, 000.00 many times will each swimmer complete in this contest? a. 10 times c. 5 times 13. Write this ratio in its simplest form: 3 dm to 20 cm. b. 16 times d. 4 times a. 20:3 b. 20:30 c. 3:2 d. 3:20

LET Review 2012

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14. Which of the following is CLOSEST to the square root of 26. Which of the following gives the sum of the polynomials 4000? (a2 + b2 + ab) and (3a2 + 4ab – 2b2)? a. 63 b. 200 c. 22 d. 19 a. 4a2 + b2 + 5ab c. 3a2 + 5ab – 2b2 b. 5a2 + 5b2 + 5ab d. 4a2 + 5ab – b2 15. The arithmetic mean of a set of 50 numbers is 38. If two numbers 45 and 35 are discarded, the mean of the 27. Of 50 students enrolled in the subject, Curriculum and remaining set of numbers is: Instruction, 90% took the final examination at the end of a. 37.24 b. 37.92 c. 37.0 d. 36.5 the term. Two-thirds of those who took the final examination passed. How many students passed the 16. If 5x>0 which of the following must be TRUE? exam? I. x <0 II. 1/x>0 III. –x<0 a. 33 b. 45 c. 34 d. 30 a. II only

b. II and III

c. I and II

d. I only

28. A cube is a rectangular solid, the length, width and height of which have the same measure called the edge (e) of the 17. If P is a positive integer in the equation 12p = q, then q cube. The volume of the cube is found by “cubing” the must be a: measure of the edge. What is the volume of the cube a. positive even integer c. positive odd integer whose edge is 4 cm? b. negative even integer d. negative odd integer a. 27 cm3 b. 64 cm3 c. 206 cm3 d. 125 cm3 18. What is the range of the following: 86, 70, 83, 90, 85, 78, 79, 81, 87 a. 12 b. 15 c. 16 d. 21

29. If a function is defined by the set of ordered pairs (1,2), (2,4), (3,8), (4,16), (5,Y), then the value of Y is: a. 10 b. 32 c. 20 d. 25

19. Change the following percents to decimals: 23%, 5%, 3%, 30. If n m +b c. n = a a. 2.3, 500, 0.30, 30.5 c. 0.23, 0.05, 0.03, 0.035 b. m = b d. n + a < m + b b. 0.23, 50, 03, 3.05 d. 0.023, 5.00, 3.0, 03.5 20. If the scores of 10 students are: 76, 80, 75, 83, 80, 79, 85, LET SEPTEMBER 2009 80, 88, 90, the mode is: a. 79 b. 85 c. 80 d. 88 1. How much greater is the sum of the first 20 counting numbers than the sum of the first 10 counting numbers? 21. A vehicle consumes one liter of gasoline to travel 10 a. 110 b. 55 c. 100 d. 155 kilometers. After a tune-up, it travels 15% farther on one liter. To the nearest tenth, how many liters of gasoline will 2. Which of these statements is always TRUE? it take for the vehicle to travel 230 kilometers? a. 23 liters b. 20 liters c. 23.15 liters d. 20.15 liters a. The sum of 3 consecutive pages of a book is always odd. b. The sum of two square numbers CANNOT be odd 22. How many thousand are there in one million? c. The sum of the page number of two consecutive pages a. 100 b. 500 c. 50 d. 1000 of a book is even d. The sum of 5 consecutive numbers is always divisible 23. How many twenty thousands are there in one million? by 5 a. 500 b. 50 c. 100 d. 1000 3. A father and son working together can finish painting a 24. Which of the following is the numerical form for “twenty room in 6 hours. Working alone, the father takes 9 hours to thousand twenty”? do the painting. How many painting hours will it take the a. 2,020 b. 2,000,020 c. 20,020 d. 20,000,20 son, working alone, to finish painting the room? a. 18 b. 15 c. 75 d. 12 25. A rectangle has sides of 10 and 12 units. How can the area of a square be computed if it has the same perimeter 4. Which common fraction is equivalent to 0.216? as the rectangle? a. 53/250 b. 27/125 c. 21/50 d. 108/375 a. Add 10 and 12, double the sum, then multiply by 4 b. Add 10 and 12, double the sum, divide by 4, then 5. From a 72 m bolt/roll of clothing material, 28 school multiply by 4 uniforms are made. Each uniform uses 2-m. How many c. Add 10 and 12, double the sum, divide by 4, then meters of material are left? multiply by 4 a. 8 and 3/4 c. 16 d. Add 10 and 12, double the sum, divide by 4, then square b. 10 d. 12 the quotient LET Review 2012

Focus: Mathematics

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6. Nelia takes ¾ hour to dress and get ready for school. It 17. If the width of a rectangle is reduced by 20% and the takes 4/5 hour to reach the school. If her class starts length is also reduced by 20%, what percent of the original promptly at 8:15 am; what is the latest time she can jump area is the new area of the rectangle? out of bed in order not to be late for school? a. 60% b. 64% c. 80% d. 36% a. 6:42 am b. 6:50 am c. 6:45 am d. 7:22 am 18. How many square inches are in 1 square yard? 7. In a playground for Kindergarten kids, 18 children are a. 900 b. 144 c. 1296 d. 648 riding tricycles or bicycles. If there are 43 wheels in all, how many tricycles are there? 19. Which of these has the longest perimeter? a. 8 b. 9 c. 7 d. 11 a. A square 21 cm on a side b. A rectangle 19 cm long and 24 cm wide c. An equilateral triangle whose side is 28 cm 8. Simplify 5√ - 4√ d. A right triangle whose two legs are 24 and 32 cm a. 13√ b. √ c. 33√ d. 17√ 9. A child can be admitted to Grade I if he/she is at least 5 20. Determine the midpoint of the line segment joining the points (5, -3) and (-1, 6). years and 10 months old by June of the school year. Which a. (2, 3/2) b. (2, -3/2) c. (3, 3/2) d. (1, 5/2) of these birthdates will disqualify a child from being admitted to Grade I on June 15, 2009? 21. The legs of one right triangle are 9 and 12, while those of a. June 10, 2003 c. May 5, 2003 another right triangle are 12 and 16. How much longer is b. Jan. 10, 2003 d. April 30, 2003 the perimeter of the larger triangle than the perimeter of the smaller triangle? 10. A time deposit in bank will mature after 91 days. If the a. 84 b. 7 c. 12 d. 14 deposit was made on April 28, on what date will it mature? a. June 28

22. A student has ten posters to pin up on the walls of her room, but there is space for seven. In how many ways can she choose the posters to be pinned up? 11. One side of a 45° - 45° - 90° triangle measures 5 cm. What a. 5,040 b. 604,800 c. 120 d. 7 is the length of its hypotenuse? a. 5√ cm b. 5 cm c. (5√ )/2 cm d. 5√ cm 23. One angle of a parallelogram is 35. What are the measures of the three other angles? 12. An online shop sells a certain calculator for P850 and a. 145 , 35 , 145 c. 85 , 135 , 140 charges P250 for shipping within Manila, regardless of the b. 45 , 65 , 170 d. 35 , 65 , 65 number of calculators ordered. Which of the following equations shows the total cost (y) of an order as a 24. A series of a square figures are made with match sticks. If the first three figures are the following, how many function of the number of calculators ordered (x)? matchsticks will be needed to form the sixth figure? a. y = (850 + 250)x c. y = 850x + 250 b. y = 250x +850 d. y = 850x + 250 b. July 18

c. July 28

d. July 29

13. How many glasses of Cola each to be filled with 150 cu cm of liquid can be made from 5 family size bottles of cola each containing 1.5 liters? a. 60 b. 5 c. 84 d. 40 a. 60 b. 45 c. 40 d. 50 14. A retailer buys candies for P37.50. The pack has 25 25. If 1/x = a/b then x equals the ______________ pieces of candies. If she sells each candy for P2.25, how a. product of a and b c. difference of a and b much profit does she make? b. sum of a and b d. quotient of b and a a. P15.50 b. P56.25 c. P37.50 d. P18.75 15. The edges of a cubical frame are made from plastic straws. How much longer is the total length of the plastic edges of a cube whose edge is 10 cm compared to a cube whose edge is 8 cm? a. 32 cm b. 8 cm c. 24 cm d. 16 cm

26. On a certain day, three computer technicians took turns in manning a 24-hour internet shop. The number of hours Cesar, Bert, and Danny were on duty was in the ratio 1:2:3, respectively. The shop owner pays them P40 per hour. How much would Danny receive for that day? a. P 230 b. P960 c. P160 d. P480

16. How many different rectangles can be found in the 27. Ruben’s grades in 6 subjects are 88, 89, 87, 90, 91 and diagram below? 86? What is the least grade that should aim for in the 7th subject if he has to have an average of 88? a. 91 b. 90 c. 93 d. 92 a. 8 b. 4 c. 5 d. 9 LET Review 2012

Focus: Mathematics

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LET SEPTEMBER 2008 16. 259:37 as _______________. 1. In a certain school, the ratio of boys to girls is 3 is to 7. If a. 84:12 b. 25:4 c. 5:1 d. 63:441 there are 150 boys and girls in the school, how many boys are there? 17. Which common fraction is equivalent to 0.216? a. 105 b. 90 c. 45 d. 75 a. 53/50 b. 27/125 c. 21/50 d. 108/375 2. The edges of a rectangular solid have these measures: 1.5 18. What value of x will make this proportion correct? feet by ½ feet by 3 inches. What is its volume in cubic 13 : x = 3.25 : 8 inches? a. 2 b. 3.5 c. 32 d. 20 a. 324 b. 225 c. 272 d. 27 19. How much greater is the sum of the first 20 counting 3. A water tank contains 8 liters when it is 20% full. How numbers greater than the sum of the first 10 counting many liters does it contain when 75% full? numbers? a. 60 b. 30 c. 58 d. 15 a. 110 b. 55 c. 155 d. 100 4. Which of the following has the largest value? a. 75 b. 39 c. 64 d. 93 5. How many vertices does a rectangular solid should have? a. 8 b. 6 c. 4 d. 12 6. Which of these three fractions has the largest value? a. 4/15 b. 3/4 c. 12/24 d. 5/9 7. if x = 3 and y = 2, then 2x + 3y = _____________. a. 12 b. 10 c. 14 d. 5 8. Which of these has a value different from the other three? a. 15% b. 3/20 c. 0.15 d. 12/75

20. Ana and Beth do a job together in two hours. Working alone, Ana does the job in 5 hours. How long will it take Beth to do the job alone? a. 3 and 1/3 hours c. 3 hours b. 2 and 1/3 hours d. 2 hours 21. What is the greatest common factor of the number 48, 12, and 32? a. 2 b. 4 c. 3 d. 6 22. In the following number: 5672.13498, which digit is in the thousandths place? a. 6 b. 4 c. 3 d. 9

9. The vertex angle of an isosceles triangle is 30°. What is the 23. By how much is 35% of P220 greater than 55% of P140? measure of one of the base angles? a. P10.00 b. P0.00 c. P20.00 d. P16.00 a. 150° b. 60° c. 75° d. 70° 24. Between January 2008 and August 2008, the price of gasoline rose from 28.40 to 52.90 per liter. What is the 10. What is the value of 12 2 7 19 ? percent increase over the 8-month period? a. 21 b. 22 c. 21 d. 21 a. 97.5% b. 86.27% c. 24.5% d. 46.81% 11. Joseph is 10 years older than his sister. If Joseph was 25 25. The area of a rectangle is (x2 +x-12). If its length is x + 4, years of age in 1983, in what year could he have been what is its width? born? a. x + 2 b. x - 3 c. x + 1 d. x + 6 a. 1948 b. 1963 c. 1958 d. 1953 12. Which number is wrong in this sequence? What should 26. In a Mathematics test, 35 students passed and 30% the number be for the sequence to be CORRECT? 5, 15, 16, failed. How many students took the test? 48, 144, 147, 148, 444 … a. 60 b. 50 c. 40 d. 30 a. 48; it should be 18 c. 144; it should be 132 b. 147; it should be 147 d. 144; it should be 49 27. At what rate per annum should P2400 be invested so that it will earn an interest of P880 in 8 years? 13. What number is next in this sequence: 11120, 11260, a. 6 % b. 5 % c. 4.6% d. 6% 11420, 11600? a. 11820 b. 11760 c. 11720 d. 11800 28. Each bag of rice contains 50 kg. What would be the increase per kilogram in the price if the cost of one bag of 14. Which decimal is equal to 3/100 + 2/10 + 5/1000? rice goes from P 950.00 to P 1,200.00? a. 0.532 b. 0.0523 c. 0.325 d. 0.235 a. P250.00 b. P7.50 c. P3.75 d. P5.00 15. If the counting numbers 1 to 30 are written down, how many times does the digit 2 appear? a. 11 b. 14 c. 12 d. 13

LET Review 2012

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LET APRIL 2008 1. Three brothers inherited a cash amount of P120, 000 and 14. All the seats in a bus are occupied and six persons are they divided it among themselves in the ratio of 5:2:1. standing. At the next bus stop, 13 persons got off and 5 How much more is the largest share than the smallest got in. How many seats were empty after this stop if share? everyone had a seat? a. P75,000 b. P30,000 c. P15,000 d. P60,000 a. 24 b. 23 c. 0 d. 2 2. In this number, 2457.13689, which digit is in the 15. A,B, and C are consecutive numbers. If A>B>C, what is the thousandths place? value of (A-B)(A-C)(B-C)? a. 8 b. 6 c. 3 d. 4 a. -2 b. 2 c. 1 d. -1 3. Joan bought 120 handkerchiefs at 10 pesos each. Then she 16. Which of the following could be a factor of n(n+1) if n is a sold them at 3 handkerchiefs for P50. If she sold all the positive integer less than 3? handkerchiefs, how much profit did she make? a. 8 b. 3 c. 9 d. 5 a. P400 b. P800 c. P733 d. P170 17. What number subtracted from each of 71 and 58 will 4. If the area of one circle is twice of another circle, what is result in two perfect squares? the ratio in percent of the smaller to the larger circle? a. 22 b. 42 c. 35 d. 33 a. 70% b. 25% c. 75% d. 50% 18. A recipe which is good for 4 persons calls for 2/3 cup of 5. The average of 5 different counting numbers is 10. What is milk. How much milk will be needed by a recipe for 6 the highest possible value that one of the numbers can persons? have? a. 4 cups b. 1 cup c. 2 cups d. 8 cups a. 20 b. 40 c. 30 d. 38 19. Which of the following has the greatest value: 6. What value of x will satisfy the equation: a. 2 + 22 + (2 + 2)2 c. [(2 + 2)2]2 3 0.2(2x + 1470) = x? b. 4 d. (2 + 2 + 2)2 a. 490 b. 2,130 c. 1470 d. 560 20. How many numbers of Set A are factors of any numbers 7. Which of the following is a factor of the quadratic equation of Set B? 2 x – 2x – 24 = 0? Set A = {0,1,2,3,4,5} Set B = {1,2,7,9,10} a. x - 4 b. x + 2 c. x + 6 d. x + 4 a. 6 b. 5 c. 4 d. 3 8. Julie spent one-sixth of her money in one store. In the next store, she spent three times as much as she spent in the first 21. A store lost P252,000 last year. What was its average loss store, and had 80 pesos left. How much money did she have per month? at the start? a. P20,000 b. P20,500 c. P21,500 d. P21,000 a. 240 pesos b. 252 pesos c. 300 pesos d. 360 pesos 22. If x is an odd integer and y is an even integer, which of 9. What part of an hour had passed from 2:48 pm to 3:20 the following is an odd integer? pm? a. 2x-y b. x2+y-1 c. x2 + 3y d. x – 1 a. 7/8 b. 1/3 c. 8/15 d. 8/2.5 23. The first 5 numbers in a sequence are 5,6,8,11 and 15. 10. What is 3m + 28 dm when converted to centimeters? What are the 8th and 10th numbers in the sequence? a. 480 b. 4800 c. 5800 d. 580 a. 32 and 49 b. 26 and 49 c. 27 and 42 d. 33 and 50 11. A street vendor sells 3 kg roasted peanuts at P5 per 24. What percent of 4 is 3/5 of 8? packet of 25 g which she bought for P80 per kilo. She a. 83 1/3 % b. 48% c. 80% d. 120% spent P20 for oil, fuel, and plastic bags. What is her net gain from selling all the peanuts? 25. What is the counting number that is less than 15 and a. P480 b. P380 c. P340 d. P400 when divided by 3 has a remainder of 1, but when divided by 4 has a remainder of 2? 2 12. If x = -5, what is the value of (x – 9)/(x + 3)? a. 5 b. 12 c. 8 d. 10 a. 7 b. -6 c. -8 d. 5 26. Which of these weights is heaviest? 13. What is the missing terms in the series a. 2250 g b. 2.5 kg c. 5 pounds d. 4200 mg 5,10,20,____,80,____,320? a. 50,210 b. 40,160 c. 35,135 d. 40,120 LET Review 2012

Focus: Mathematics

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27. Which of the following is the factorization of the binomial 32. The area of a square is 32x. Which of the following could x2 - 42? be an exact value for x? a. (x + 4)(x + 2) c. x(x + 2x + 2) a. 2 b. 6 c. 3 d. 4 2 b. (x – 4) d. (x – 4)(x + 4) 33. How much is 80% of 37% of 2.4? 28. What percent of 560 is 35? a. 7.2 b. 1.92 c. 19.2 d. 0.71 a. 6.5% b. 16% c. 6.25% d. 65% 34. Two buses leave the same station at 9:00 pm. One bus 29. How much interest would be paid on a bank loan of travels north at the rate of 30 kph and the other travels P 30,000 for 8 months at 12% annual interest? east at 40 kph. How many kilometers apart are the buses a. P3,240 b. P2,400 c. P2,800 d. P3,600 at 10 pm? a. 140 km b. 100 km c. 70 km d. 50 km 30. Which of the fractions has the LEAST value? a. 7/12 b. 8/9 c. 6/17 d. 7/8 31. What is the average of ½, ¼, and 1/3? a. 13/12 b. 13/24 c. 13/23

LET Review 2012

Focus: Mathematics

d. 13/29

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Answer Sheet (Mathematics) Name:____________________________________________________ Date Taken: _____________________________________________

____ 1. ____ 2. ____ 3. ____ 4. ____ 5. ____ 6. ____ 7. ____ 8.

____ 9. ____ 10. ____ 11. ____ 12. ____ 13. ____ 14. ____ 15. ____ 16.

Total Score: _________________________________________

30

Partial Score: ____ 17. ____ 18. ____ 19. ____ 20. ____ 21. ____ 22. ____ 23. ____ 24.

____ 25. ____ 26. ____ 27. ____ 28. ____ 29. ____ 30.

Partial Score:

27 ____ 1. ____ 2. ____ 3. ____ 4. ____ 5. ____ 6. ____ 7.

____ 8. ____ 9. ____ 10. ____ 11. ____ 12. ____ 13. ____ 14.

____ 15. ____ 16. ____ 17. ____ 18. ____ 19. ____ 20. ____ 21.

____ 22. ____ 23. ____ 24. ____ 25. ____ 26. ____ 27.

Partial Score:

28 ____ 1. ____ 2. ____ 3. ____ 4. ____ 5. ____ 6. ____ 7.

____ 8. ____ 9. ____ 10. ____ 11. ____ 12. ____ 13. ____ 14.

____ 15. ____ 16. ____ 17. ____ 18. ____ 19. ____ 20. ____ 21.

____ 22. ____ 23. ____ 24. ____ 25. ____ 26. ____ 27. ____ 28.

Partial Score:

34 ____ 1. ____ 2. ____ 3. ____ 4. ____ 5. ____ 6. ____ 7. ____ 8. ____ 9.

LET Review 2012

Focus: Mathematics

____ 10. ____ 11. ____ 12. ____ 13. ____ 14. ____ 15. ____ 16. ____ 17. ____ 18.

____ 19. ____ 20. ____ 21. ____ 22. ____ 23. ____ 24. ____ 25. ____ 26. ____ 27.

____ 28. ____ 29. ____ 30. ____ 31. ____ 32. ____ 33. ____ 34.

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Answer Sheet (Mathematics - ANSWERS) Name:____________________________________________________ Date Taken: _____________________________________________

_____ 1. A ____ 2. C ____ 3. A ____ 4. C ____ 5. D ____ 6. B ____ 7. D ____ 8. B

____ 9. D ____ 10. D ____ 11. C ____ 12. B ____ 13. C ____ 14. A ____ 15. B ____ 16. B

Total Score: _________________________________________

30

Partial Score: ____ 17. A ____ 18. A ____ 19. D ____ 20. C ____ 21. B ____ 22. D ____ 23. B ____ 24. C

____ 25. D ____ 26. D ____ 27. D ____ 28. B ____ 29. B ____ 30. D

Partial Score:

27 ____ 1. D ____ 2. D ____ 3. A ____ 4. B ____ 5. C ____ 6. A ____ 7. C

____ 8. D ____ 9. B ____ 10. C ____ 11. D ____ 12. C ____ 13. D ____ 14. D

____ 15. C ____ 16. D ____ 17. B ____ 18. C ____ 19. D ____ 20. A ____ 21. C

____ 22. B ____ 23. A ____ 24. D ____ 25. D ____ 26. D ____ 27. A

Partial Score:

28 ____ 1. A ____ 2. A ____ 3. B ____ 4. B ____ 5. A ____ 6. B ____ 7. A

____ 8. D ____ 9. C ____ 10. C ____ 11. C ____ 12. D ____ 13. D ____ 14. D

____ 15. D ____ 16. A ____ 17. B ____ 18. C ____ 19. C ____ 20. A ____ 21. B

____ 22. B ____ 23. B ____ 24. B ____ 25. B ____ 26. B ____ 27.C ____ 28. D

Partial Score:

34 ____ 1. D ____ 2. B ____ 3. B ____ 4. D ____ 5. B ____ 6. A ____ 7. D ____ 8. A ____ 9. C

LET Review 2012

Focus: Mathematics

____ 10. D ____ 11. C ____ 12. C ____ 13. B ____ 14. D ____ 15. A ____ 16. B ____ 17. A ____ 18. B

____ 19. C ____ 20. C ____ 21. D ____ 22. C ____ 23. D ____ 24. D ____ 25. D ____ 26. B ____ 27. D

____ 28. C ____ 29. B ____ 30. C ____ 31. C ____ 32. A ____ 33. D ____ 34. D

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Let Math April

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