Multiplying & dividing fractions & decimals
Multiplying & dividing unit fractions
Multiplying a fraction by a whole number is just repeated adding. Seeing that makes the rule obvious rather than something to memorise.
Multiplying is repeated adding
1/6 × 4 means four lots of 1/6. 1/6 + 1/6 + 1/6 + 1/6 = 4/6 So you multiply the NUMERATOR by the whole number and leave the denominator alone. The pieces stay the same size; you just have more of them.
The common mistake
Arun writes 1/6 × 5 = 5/30. That is wrong. He multiplied the bottom as well as the top. But five lots of one-sixth cannot be SMALLER than one-sixth — and 5/30 is exactly 1/6. The denominator tells you the size of each piece, and that does not change when you take more of them. The answer is 5/6.
Simplifying the answer
1/6 × 4 = 4/6, and 4/6 can be cancelled. Both 4 and 6 divide by 2, giving 2/3. Always check whether the answer will cancel.
Dividing a fraction by a whole number
1/2 ÷ 3 means splitting a half into 3 equal parts. Each part is 1/6 of the whole. So 1/2 ÷ 3 = 1/6. Here you multiply the DENOMINATOR by the whole number, because the pieces are getting smaller. Dividing makes each piece smaller; multiplying gives you more pieces of the same size.
Worked examples
1/6 × 4
- Four lots of one sixth.
- 1/6 + 1/6 + 1/6 + 1/6 = 4/6.
- Simplify: both divide by 2.
Answer: 1/6 × 4 = 2/3
1/5 × 6
- Six lots of one fifth.
- Multiply the numerator: 1 × 6 = 6.
- The denominator stays 5.
Answer: 1/5 × 6 = 6/5, or 1 1/5
Correcting Arun
- He wrote 1/6 × 5 = 5/30.
- 5/30 simplifies to 1/6 — the same as he started with.
- Only the numerator should be multiplied.
Answer: 1/6 × 5 = 5/6
1/2 ÷ 3
- Split one half into 3 equal parts.
- Each part is one sixth of the whole.
Answer: 1/2 ÷ 3 = 1/6
1/4 ÷ 2
- Split a quarter into 2 equal parts.
- Multiply the denominator: 4 × 2 = 8.
Answer: 1/4 ÷ 2 = 1/8
Where you see this in real life
- Four people each taking 1/6 of a cake between them.
- Sharing half a pizza between three people.
- Working out how much ribbon you need for 5 pieces each 1/4 m long.
Key words
- Unit fraction
- A fraction with 1 as its numerator.
- Numerator
- The top number of a fraction.
- Denominator
- The bottom number of a fraction.
- Simplify
- Cancel a fraction to its smallest form.
- Improper fraction
- A fraction whose top is bigger than its bottom.
- Repeated addition
- Adding the same amount several times.