Multiplication and division (2)
Multiplying & dividing fractions
In this unit you learn to multiply a proper fraction by a whole number and to divide a proper fraction by a whole number. Always estimate the size of your answer first, and change improper fractions to mixed numbers.
Multiplying a fraction by a whole number
To multiply a proper fraction by a whole number, multiply the numerator by the whole number and keep the same denominator. Example: 2/5 × 3. Multiply the numerator: 2 × 3 = 6, keep the denominator 5, so 2/5 × 3 = 6/5. Repeated addition is the same as multiplying: 2/5 + 2/5 + 2/5 = 3 × 2/5.
Writing the answer as a mixed number
If the numerator becomes larger than the denominator, you have an improper fraction. Change it to a mixed number by working out how many whole ones fit in. Example: 6/5 = 1 1/5. Some answers can also be written in a simpler form, such as 6/9 = 2/3.
Dividing a fraction by a whole number
To divide a proper fraction by a whole number, split each part of the fraction into that many equal pieces. This multiplies the denominator by the whole number. Example: 2/3 ÷ 4. Split each third into 4 equal parts, giving 3 × 4 = 12 parts. The 2 thirds become 2 parts out of 12: 2/3 ÷ 4 = 2/12 = 1/6.
Dividing is the same as multiplying by a unit fraction
Dividing by a whole number is the same as multiplying by its unit fraction. Dividing by 2 is the same as multiplying by 1/2, dividing by 3 is the same as multiplying by 1/3, and so on. Example: 1/2 ÷ 3 = 1/2 × 1/3 = 1/6.
Estimate first
Before you calculate, estimate the size of the answer so you know what to expect. For 5 × 2/3, five lots of about two-thirds is a little more than 3, so the answer should be between 3 and 4. Working it out gives 10/3 = 3 1/3.
Worked examples
Multiply a proper fraction by a whole number
- Work out 2/5 × 3.
- Multiply the numerator by the whole number: 2 × 3 = 6.
- Keep the denominator the same: 5.
- 6/5 is improper, so change it to a mixed number.
Answer: 2/5 × 3 = 6/5 = 1 1/5
Divide a proper fraction by a whole number
- Work out 2/3 ÷ 4.
- Draw a bar showing thirds and split each third into 4 equal parts.
- There are now 3 × 4 = 12 equal parts, and 2 of them are shaded: 2/12.
- Simplify the fraction.
Answer: 2/3 ÷ 4 = 2/12 = 1/6
Dividing as multiplying by a unit fraction
- Work out 1/2 ÷ 3.
- Dividing by 3 is the same as multiplying by 1/3.
- Multiply the denominators: 2 × 3 = 6.
Answer: 1/2 ÷ 3 = 1/2 × 1/3 = 1/6
Estimate the size of the answer
- Before calculating 5 × 2/3, estimate its size.
- Five lots of about 2/3 is a little more than 3, so expect an answer between 3 and 4.
- Now calculate: 5 × 2 = 10 over 3, giving 10/3.
Answer: 5 × 2/3 = 10/3 = 3 1/3, which is between 3 and 4
Where you see this in real life
- Sharing half a cake equally between 4 friends, so each gets 1/8.
- Three people splitting 3/4 of a pie equally between them.
- Working out how much pizza each child gets when a fraction is shared out.
Key words
- Numerator
- The top number of a fraction — how many parts you have.
- Denominator
- The bottom number of a fraction — how many equal parts the whole is split into.
- Proper fraction
- A fraction where the numerator is smaller than the denominator, such as 2/3.
- Unit fraction
- A fraction with a numerator of 1, such as 1/3.
- Improper fraction
- A fraction where the numerator is bigger than the denominator, such as 6/5.
- Mixed number
- A whole number and a proper fraction together, such as 1 1/5.
- Operator
- Multiplying or dividing that acts on a fraction to change it.