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Form 2 [CHAPTER 4: FRACTION AND DECIMALS]

C.Camenzuli | www.smcmaths.webs.com 1

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

Denominator Numerator

Form 2 [CHAPTER 4: FRACTION AND DECIMALS]

C.Camenzuli | www.smcmaths.webs.com 2

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]

C.Camenzuli | www.smcmaths.webs.com 3

(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]

C.Camenzuli | www.smcmaths.webs.com 4

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]

C.Camenzuli | www.smcmaths.webs.com 5

Step 2: Multiply top and bottom so as to get the same denominator (Equivalent fractions). 5

5

1 53 15

=

uuuuuuruuuuuurand

3

3

2 65 15

=

uuuuuuuruuuuuuur

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]

C.Camenzuli | www.smcmaths.webs.com 6

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]

C.Camenzuli | www.smcmaths.webs.com 7

= 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]

C.Camenzuli | www.smcmaths.webs.com 8

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]

C.Camenzuli | www.smcmaths.webs.com 9

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]

C.Camenzuli | www.smcmaths.webs.com 10

Example 4: Find 2 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]

C.Camenzuli | www.smcmaths.webs.com 11

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]

C.Camenzuli | www.smcmaths.webs.com 12

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]

C.Camenzuli | www.smcmaths.webs.com 13

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]

C.Camenzuli | www.smcmaths.webs.com 14

(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]

C.Camenzuli | www.smcmaths.webs.com 15

(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]

C.Camenzuli | www.smcmaths.webs.com 16

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.