Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Chapter 4: Fractions and Decimals 4.1 – Understanding Fractions (Revision of equivalent fractions) A fraction describes part of a whole. Each fraction consists of a
denominator (bottom) and a numerator (top), representing
(respectively) the number of equal parts that an object is divided
into.
Equivalent fractions
Equivalent fractions are fractions that are equal.
Are the following equivalent fraction?
(i) 25
, 820
(ii) 32
, 63
(iii) 56
, 2530
⅜ Denominator Numerator
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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4.2 − Adding and subtracting fractions and mixed numbers Example 1: Find: 2 1
3 5+
Step 1: Find the lowest common multiple of both denominators. The multiples of 3 are 3, 6, 9, 12, 15, … The multiples of 5 are 5, 10, 15, … 15 is the lowest common multiple (LCM) Step 2: Change the denominators to make them both equal to the LCM by using equivalent fractions. 2 103 15= (Multiplying top and bottom by 3)
1 35 15= (Multiplying top and bottom by 3)
Step 3: Add or subtract both fractions with the same denominator.
So 2 1 10 3 133 5 15 15 15+ = + =
Exercise 1: Work out the following:
(i) 5 36 4−
(ii) 2 13 5−
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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(iii) 1 13 6+
(iv) 3 14 3+
Adding and Subtracting Mixed Numbers Example 2: Work out: 1 12 5
3 2+
Step 1: Add the whole numbers: 2 + 5 = 7
Step 2: Work out 1 13 2+
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Exercise 2: Work out:
(i) 1 27 34 3−
(ii) 3 62 54 7+
4.3 – Ordering fractions Example 1: Which fraction is bigger?
a) 1 23 5or
Step1: Find the LCM The multiples of 3 are 3, 6, 9, 12, 15, … The multiples of 5 are 5, 10, 15, … 15 is the lowest common multiple (LCM)
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Step 2: Multiply top and bottom so as to get the same denominator (Equivalent fractions). 5
5
1 53 15
×
×
=
uuuuuur
uuuuuurand
3
3
2 65 15
×
×
=
uuuuuuur
uuuuuuur
Step 3: Compare the numerator. 25
is bigger than 13
.
Example 1: Which fraction is bigger? 3 4
4 5or
Exercise 2: Order the following
34 ′25 ′
610 ′
12
Step 1: Find the LCM of the denominators The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, … The multiples of 4 are 4, 8, 12, 16, 20, … The multiples of 5 are 5, 10, 15, 20, … The multiples of 10 are 10, 20, … 20 is the lowest common multiple (LCM) Step 2: Get all fractions with a common denominator
34 =
1520
25 =
820
610 =
1220
12 =
1020
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Step 2: Compare the numerators (in ascending order)
25 ′12 ′
610 ′
34
Example 2: Arrange in descending order: 2 11 23 7 3, , , ,3 15 30 10 5
4.4 – Multiplying fractions To multiply a fraction by an integer, multiply the numerator of the fraction by the integer. Do not change the denominator of the fraction.
Example 1: Work out: 263
×
2 6 263 3
×× =
2
1
6 23×
=
2 21×
=
= 4 The answer is 4. To multiply two fractions, multiply the numerators and then multiply the denominators.
Example 2: Work out: 3 24 3×
= 3 24 3×
×
=1 1
2 1
3 24 3×
×
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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= 1 12 1×
×
12
=
Example 4: Work out: 2 333×
Example 5: Work out: 2 23 5×
Example 6: Work out: 5 714 10×
Example 7: Work out: 4 123 18of
When multiplying mixed numbers, first write the mixed numbers as improper fractions.
Example 8: 2 42 13 5×
Step 1: Converting the mixed numbers as improper fractions: 2 823 3= (2 x 3 = 6 + 2 = 8)
4 915 5= (1 x 5 = 5 + 4 = 9)
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Step 2: Carry out the multiplication:
Therefore, 2 4 8 92 13 5 3 5× = ×
3
1
8 93 5×
=×
8 31 5×
=×
245
=
Step 3: Change the improper faction into a mixed number:
24 445 5
= =
Example 9: Work out 1 32 12 5×
Example 10: Work out 1 6 5
3 15 7× ×
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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4.5 – Fraction of a quantity Example 1: Find 3
5 of 25 cm.
To work out this problem we need to multiply 35
by 25cm.
To find a fraction of an amount, multiply by the numerator and divide by the denominator:
3 3 25255 5
×× =
5
1
3 255×
=
3 51×
=
= 15 cm Answer: 15 cm
Example 2: Find of 63 g.
Answer
Example 3: Find of €30
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Example 4: Find 2⅓ of 66 4.6 – Dividing fractions Example 1: Work out 4 3
5÷
Step 1: Find the reciprocal of the divisor.
The reciprocal of 3 is 13
Step 2: Multiply the dividend by the reciprocal
4 15 3×
4 15 3×
=×
415
=
Answer: 415
Example 2: Work out 5 36 4÷
Step 1: Find the reciprocal of the divisor.
The reciprocal of 34
is 43
Step 2: Multiply the dividend by the reciprocal 5 46 3×
2
3
5 46 3×
=×
5 23 3×
=×
109
=
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Answer: 10 119 9=
Example 3: Work out 5 158 32÷
Division of Mixed numbers
Example 4: Work out 4 12 25 10÷
Step 1: Convert the mixed numbers to improper fractions.
4 1425 51 21210 10
=
=
Thus, 4 1 14 212 25 10 5 10÷ = ÷
Step 2: Find the reciprocal of the divisor
The reciprocal of 2110
is 1021
Step 3: Multiply the dividend by the reciprocal
14 105 21× 2 2
31
14 105 21×
=×
2 21 3×
=×
43
=
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Answer: 4 113 3=
Example 5: Work out 5 15 16 9÷
Example 6: Work out 2 1 14 79 3 5× ÷
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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4.7 – Changing from fraction to decimal All fractions can be changed back into a decimal
Method 1: Using equivalent fractions
Step 1: Check the denominator
Step 2: Create an equivalent fraction with denominators 10, 100, 1000 etc.
Step 3: Convert into a decimal
Example1: Convert 25
into a decimal
1) Denominator is 5
2) Equivalent fraction
2 45 10=
3) Convert into a decimal
0.4
Example 2: Convert the following fractions into decimals
(i) 45
(ii) 720
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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(iii) 325
(iv) 850
Method 2: Short division
It is not always possible to create an equivalent fraction with denominators being multiples of
10. When this is not possible we can either use the calculator or perform a short division.
Example 1: Use short division to change these fractions to decimals.
(i) 38
(ii) 14
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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(iii) 16
4.8 – Changing from decimal to fraction A terminating decimal is a decimal which ends.
E.g. 0.26, 0.628 are terminating decimals
All terminating decimals can be converted into fractions.
Step 1: Observe the decimal
Step 2: Find the place value of the digit further to the right
Step 3: The place value of the last digit shows the number over which we have to
express the fraction
Step 4: Simplify the resulting fraction
Example 1: Convert 0.24 into a fraction
1) Observe the decimal
0.24
2) Finding the place value
The place value of 4 is a hundredth( 1100
)
3) Expressing the fraction
240.24100
=
4) Simplify
24 12 60.24100 50 25
= = =
Form 2 [CHAPTER 4: FRACTION AND DECIMALS]
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Example 2: Convert 2.5 into a fraction
5 12.5 2 210 2
= =
Example 3: Convert the following decimals into fraction
(i) 5.45
(ii) 0.67
(iii) 56.42
(iv) 0.0003