MATHEMATICS - FORM 1
DECIMALS
DECIMAL AND FRACTION
1. A decimal is a fraction in which the denominator is 10 or a power of 10. For example :-
2. The decimal point ( the dot ) separates the whole number from its fractional part. A number written with a decimal point is known as a decimal. For example:-
3. The number of digits after the decimal point is the same as the number of zeros in the denominator. For example:-
A) Representing Decimals with Diagrams Decimals can be represented with diagrams.
For example :1 10
3 100
9 1 000
(1 out of 10) ( 3 out of 100) ( 7 out of 1 000)
B) Writing Decimals based on given Diagrams Worked Example
Write a decimal to represent the shaded area in the diagram above. Solution The shaded area is 3 . 10 Therefore, the decimal is 0. 3.
C) Conversion of Fractions into Decimals and vice versa 1. A fraction in which the denominator is 10 or a power if 10 can be converted to a decimal mentally. Worked Example Express the following as decimals. (a) 7 10 (b) 52
(c) 138 1 000
100 Solution
Worked Example Change the following into fractions with denominators of 10 or powers of 10. (a) 0.6
(b) 0.55
(c) 0.374
Solution
2. A fraction can also be converted to a decimal by dividing the numerator by the denominator. 3. Mixed numbers can also be expressed as decimals.
Worked Example Express the following fractions as decimals. (a) 3 4
(b) 7 2 5
Solution
Worked Example
Express the following as decimals. (a) 1 14 100 Solution
(b) 4 5 20
Worked Example
Express the following decimals as mixed numbers. (a) 3. 6
(b) 11. 025
Solution (a) 3. 6 = 3 + 0. 6 =3+6 10 =36 10
PLACE VALUE AND DIGIT VALUE IN DECIMALS A) Place Value and Digit Value 1. Each digit in a decimal has its own place value and digit value.
Look at the decimal 8.235 below.
Worked Example State the place value and digit value of the underlined digit in each of the follow decimals. (a) 13.46
(b) 7.095
Solution (a) Tenths ; 0.4 (b) Thousandths ; 0.005 2. Decimal places are the number of places occupied by the digits after the decimal point. For example :(a) 0.3 1 decimal place (b) 0.06
2 decimal places (c) 3.152 3 decimal places (d) 9.7420 4 decimal places
B) Representing Decimals on Number Lines A decimal can be represented on a number line. For example :-
Worked Example
State the decimal represented by each of the letters on the number line above. Solution
One part = 4.95 - 4.80 = 0.15 F = 4.95 + 0.15 + 0.15 = 5.25 G = 5.25 + 0.15 = 5.40 Therefore, F = 5.25 and G = 5.40.
C) Comparing Two Decimals When comparing two decimals, arrange the digits of the decimals according to their place values and then compare their digit values from left to right. Worked Example Which is greater, 24.853 or 24.832 Solution Arrange the decimals as shown below.
At the place value of hundredths, 5 is larger than 3. Therefore, 24.853 is greater than 24.832. Worked Example Fill in each box with ' >' or ' <'.
Solution
D) Order of Decimals We can arrange decimals in increasing or decreasing order according to their values. Worked Example Arrange 7.1, 6.4, 6.83, 6.81 in decreasing order. Solution
Worked Example
Arrange 11 , 2.73 and 2.732 in increasing order. 4 Solution
E) Rounding Off Decimals. A Decimal can be rounded off to a whole number or certain decimal places. The rules for rounding off decimals are as follows. (a) Look at the digit to the right of the digit that is to be rounded off. (b) If the digit is more than or equal to 5, add 1 to the digit that is to be rounded off and ignore the rest of the digits after it. (c) If the digit is less than 5, maintain the digit that is to be rounded off and ignore the rest of the digits after it. Worked Example
Round off 8.3752 correct to (a) 1 decimal place, (b) 2 decimal places, (c) 3 decimal places. Solution
Worked Example Round off the following decimals to the nearest whole numbers. (a) 8.6168 (b) 30.324 Solution
(c) 352.84
ADDITION AND SUBTRACTION OF DECIMALS 1. Decimals are added and subtracted in the same way as whole numbers. 2. When two decimals are added together or subtracted from each other, the decimal points must be placed directly one below the other and the digits written in the correct place value columns. A) Addition of Decimals Worked Example Calculate (a) 0.382 + 4.264 (b) 5 + 8.8 Solution
Worked Example Calculate (a) 34.2 + 26.73 + 9.985 (b) 9.25 + 54 + 17.631 Solution
B) Problem Solving involving Addition off Decimal Worked Example Salman is 1.75 m tall and Arjun is 0.19 m taller than Salman. How tall is Arjun. Solution 1. Understand the problem
Given information : Salman's height = 1.75 m Arjun is taller than Salman by 0.19 m Find : Height of Arjun 2. Devise a plan Use addition. 3. Carry out the plan Arjun's height = Salman's height + 0.19 m 1.75 m + 0.19 m = 1.94 m 1.75 +0.19 1.94 Therefore, Arjun is 1.94 m tall. 4. Check 1.94 - 1.75 0 . 1 9 ( The difference ) Worked Example Dewi collected 21.10 litres of latex on Monday, 26 litres on Tuesday and 23.906 litres on Wednesday. Find the total amount of latex collected. Solution
The total amount of latex collected was 69.906 litres. C) Subtraction of Decimal Worked Example Calculate (a) 0.85 - 0.35 (b) 83 - 26.421 Solution
Worked Example Calculate (a) 57.3 - 35.82 - 2.346 (b) 81 - 24.907 - 7.5 Solution
D) Problem Solving involving Subtraction of Decimals Worked Example Suziela weighs 52.03 kg and Azman weighs 60.4 kg. Find the difference in their body mass. Solution 1. Understand the problem Given information : Suziela's mass = 52.03 kg Azman's mass = 60.4 kg Find : Difference in their body mass 2. Devise a plan Use subtraction. 3. Carry out the plan 60.4 kg - 52.03 kg
Therefore, the difference in their body mass is 8.37 kg. 4. Check
52.03 + 8.37 60.40 Worked Example Zaiton bought 6.3 m of cloth. She used 2.15 m to make a shirt and 3.28 m to make a pair of pants. How much of the cloth remained ? Solution 6.3 m - 2.15 m - 3.28 m = 0.87 m
Therefore, 0.87 m of the cloth remained.
MULTIPLICATION AND DIVISION OF DECIMALS A) Multiplication of Decimals i - Generel multiplication of decimals 1. To find the product of decimals, multiply the numbers in the same way as for whole numbers first. Then, put in the decimal point. 2.The number of decimal places in the answer must correspond to the total number of decimal places in the decimals being multiplied. Worked Example Calculate (a) 0.73 x 0.7 Solution
ii) Multiplying a decimal by a power of 10 When multiplying a decimal by 10, 100, 1 000, ect., we move the decimal point 1, 2, 3, ect. places respectively to the right. Worked Example Calculate (a) 1.42
(c) 0.8
(b) 6.973
( d) 0.0032
Solution
2. Similarly, when we multiply a decimal by
0.1, 0.01, 0.001, ect., we move the decimal point 1, 2, 3, ect. Place respectively to the left. Worked Example Calculate (a) 24.8 x 0.1, (b) 53.62 x 0.01, (c) 21.73 x 0.001 Solution
Worked Example Calculate 0.38 x 0.7 x 0.5. Solution
B) Problem Solving involving Multiplication of Decimals Worked Example Dalina needs 0.95 kg of flour to make a cake. How much flour is needed to make 8 cakes ? Solution 1. Understand the problem Given information : Amount of flour to make 1 cake = 0.95 Find : Amount of flour to make 8 cakes 2. Devise a plan Use multiplication. 3. Carry out the plan 8 x 0.95 kg = 7.60 x
0.95 8 7.60
Therefore, 7. 60 kg of flour is needed to make 8 cakes. 4. Check
Multiply again to verify the answer. Worked Example A apple costs RM0.70. The price of a jackfruit is 4.8 times the price of the apple. What is the cost of 5 jackfruits ? Solution Cost of a apple = RM0.70 Cost of a jackfruit = 4.8 x RM0.70 Cost of 6 jackfruits = 5 x 4.8 x 0.70 = RM16.80 0.70 4.8 560 2 8 0__ 3.360 x
x
3.36 5 16.80
Therefore, the cost of 6 jackfruits is RM16.80.
C) Division of Decimals i - Dividing a whole number or a decimal by 10 or a power of 10 When a whole number or a decimal is divided by 10, 100, 1 000,... the decimal point is moved 1, 2, 3,,... places respectively to the left. Worked Example Calculate (a) 5 ÷ 10, Solution
(b) 270 ÷ 1 000.
Worked Example Calculate (a) 0.7 ÷ 10,
(c) 38.46 ÷ 1 000.
(b) 231.4 ÷ 100, Solution
ii - Dividing a whole number by a whole number Worked Example Calculate 35 ÷ 4 Solution
Therefore, 35 ÷ 4 = 8.75 Worked Example Calculate 8 ÷ 3 and give your answer correct to 4 decimal places. Solution
Therefore, 8 ÷ 3 = 2. 66666... = 2. 6667 ( 4 d.p ) iii - Dividing a decimal by a whole number and vice versa 1. Dividing a decimal by a whole number is equal sharing. Worked Example Find the value of each of the following. (a) 38.4 ÷ 4
(b) 0.22 ÷ 22 Solution
2. When dividing a whole number by a decimal, convert the divisor to a whole number first by moving its decimal point to the right. Worked Example Find the value of each of the following. (a) 6 ÷ 0.2 (b) 33 0.11 Solution
iv - Dividing a decimal by a decimal When dividing a decimal by a decimal, use the idea of equivalent fractions to convert the divisor to a whole number. Shift the decimal points the same number of places in the dividend and divisor to make the divisor a whole number. Worked Example Find the value of each of the following. (a) 5.52 ÷ 0.6 (b) 0.072 0.8 Solution
v - Dividing a decimal by a fraction and vice versa When dividing a decximal by a fraction, change the operation from division to multiplication and invert the fraction at the same time. Worked Example Find the value of each of the following. (a) 3.72 ÷ 3 4 Solution
D) Problem Solving involving Division of Decimals Worked Example Akmal buys 6 tickets for a circus show costing RM85.80. How much does each ticket cost ? Solution 1. Understand the problem Given information : Number of tickets bought = 6 Cost of 6 tickets = RM 85.80 Find : Cost of 1 tickets 2. Devise a plan Use division 3. Carry out the plan RM85.80 ÷ 6 = RM14.30
Therefore, the cost of 1 ticket is RM14.30. 4. Check x
14.30 6 85.80
COMBINED OPERATIONS OF +, -, x, ÷ OF DECIMALS A) Combined Operations of Addition and Subtraction Worked Example Find the value of each of the following. (a) 7.41 + 6.35 - 10.02 (b) 8.41 - 31 + 2.07 4 Solution
B) Problem Solving involving Addition and Subtraction Worked Example A vessel weighing 0.98 kg contains 25.8 kg of rice. If 10.5 kg of the rice is used, find the total mass of the vessel with the remaining rice. Solution 1. Understand the problem Given information : Mass of vessel = 0.98 kg
Mass of rice = 25.8 kg Mass of rice used = 10.5 kg Find : Mass of the vessel and remaining rice 2. Devise a plan Perform addition followed by subtraction. 3. Carry out the plan 0.98 + 25.8 - 10.5 = 26.78 - 10.5 = 16. 28 0.96 + 2 5 . 8__ 26.78 - 1 0 . 5__ 16.28 Therefore, the total mass of the vessel and remaining rice is 16.28 kg.
C) Combined Operations of Multiplication and Division Worked Example Calculate each of the following. (a) 4.6 x 0.8 ÷ 1.6 Solution
Worked Example Calculate each of the following. (a) 7 x 0 . 9 ÷ 4 5
Solution (a) 7 x 0 . 9 ÷ 4 5 = 6. 3 ÷ 4 ( Invert the fraction. ) 5 = 6. 3 x 5 4 = 31. 5 ÷ 4 = 7. 875
D) Problem Solving involving Multiplication and Division Worked Example A shopkeeper buys 5 sacks of sugar each weighing 64.2 kg. If he packs the sugar into plastic bags of 11 kg
2 each, how many bags are required ? Solution 1. Understand the problem Given information : Mass of a sack of sugar = 64.2 kg 5 sacks of sugar are packed into plastic bags of 11 kg. Find : Number of plastic bags required 2. Devise a plan Perform multiplication followed by division. 3. Carry out the plan
5 x 64. 2 ÷ 11 2 = 5 x 64. 2 ÷ 3 2 = 192. 6 ÷ 3 2 = 192. 6 x 2 3 = 385. 2 ÷ 3 = 128. 4 4. Check 3 x 128. 4 = 192.6 2 192. 6 ÷ 3 = 64. 2 Worked Example Amy bought 20 eggs costing RM3. 80. Lissa bought 35 of those eggs. How much did Lissa pay ? Solution Cost of 20 eggs = RM3.80 Cost of 35 eggs = 3.80 ÷ 20 x 35 = 0.19 x 35 = 6.65
Therefore, Lissa paid RM6.65.
E) Combined Operations of Addition, Subtraction, Multiplication and Division Worked Example Calculate each of the following. (a) 12. 5 - 0. 26 x 2. 6 (b) 3. 2 x 4 - 10. 4 ÷ 4 (c) 5. 12 - 2. 8 ÷ 1 4 Solution
