Maths
How do you multiply fractions?
Quick answer
To multiply fractions, multiply the numerators (tops) together and the denominators (bottoms) together, then simplify. You do not need a common denominator: 2/3 × 4/5 = 8/15.
In one sentence
To multiply fractions, multiply the tops together, multiply the bottoms together, and simplify the answer.
Explained
Multiplying fractions is the easiest of the four fraction operations. There is one rule:
numerator × numerator over denominator × denominator
So 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15. Unlike adding and subtracting, you do not need a common denominator, and it does not matter whether the denominators are the same or different. The rule is the same either way.
Why does it work? Think of the word "of". 1/2 × 3/4 means "half of three quarters". Picture a bar of chocolate split into quarters, with 3 of the 4 pieces shaded. Now cut every piece in half. The bar has 8 equal pieces, and half of the shaded part is 3 of them: 3/8. The bottoms multiplied (4 × 2 = 8 pieces) and the tops multiplied (3 × 1 = 3 pieces).
pizza
cake
pencil
An area model shows the same thing. Draw a square, shade 2/3 of it in one direction and 4/5 of it in the other. The overlap is 8 of the 15 little rectangles: 8/15.
Whole numbers. Write the whole number as a fraction over 1. 4 × 2/9 = 4/1 × 2/9 = 8/9. Only the numerator gets bigger, because you have 4 lots of 2/9.
Mixed numbers. Change them to improper fractions first (see improper fractions and mixed numbers). 2 1/2 × 1 1/5 = 5/2 × 6/5 = 30/10 = 3.
Cancel first to keep numbers small. If a top and a bottom (in either fraction) share a factor, divide both by it before you multiply. In 4/9 × 3/8, the 4 and the 8 share 4, and the 3 and the 9 share 3, leaving 1/3 × 1/2 = 1/6. You would get the same answer by working out 12/72 and then simplifying, but cancelling first is quicker. This is sometimes called cross-cancelling.
One surprise: multiplying by a fraction smaller than 1 makes a number smaller. 1/2 × 10 is 5, because taking half of something leaves less of it.
Examples
| Calculation | Multiply | Simplify |
|---|---|---|
| 1/2 × 1/3 | (1 × 1)/(2 × 3) = 1/6 | 1/6 |
| 2/3 × 3/5 | 6/15 | 2/5 |
| 3/4 × 8 | 24/4 | 6 |
| 5/6 × 3/10 | 15/60 | 1/4 |
| 7/8 × 4/7 | 28/56 | 1/2 |
| 2 1/2 × 1 1/5 | 5/2 × 6/5 = 30/10 | 3 |
Worked example: 5/6 × 3/10. Cancel 5 and 10 (÷5) to get 1 and 2; cancel 3 and 6 (÷3) to get 1 and 2. Now 1/2 × 1/2 = 1/4. Check without cancelling: 15/60 = 1/4. Same answer.
Common mistakes
- Finding a common denominator first. That is for adding and subtracting. For multiplying, go straight to top × top and bottom × bottom.
- Multiplying the whole number by the top and the bottom. 3 × 2/5 is 6/5, not 6/15. Write 3 as 3/1 to see why.
- Multiplying mixed numbers in parts. 2 1/2 × 1 1/2 is not 2 1/4. Convert first: 5/2 × 3/2 = 15/4 = 3 3/4.
- Leaving the answer unsimplified. 6/15 is correct but 2/5 is finished. Cancelling before you multiply saves this step.
Step by step
Step by step
- Make everything a fractionWrite whole numbers over 1 and change mixed numbers into improper fractions.
- Cancel if you canIf any top and any bottom share a factor, divide both by it.
- Multiply the topsThis gives the new numerator.
- Multiply the bottomsThis gives the new denominator.
- SimplifyReduce the fraction and, if it is top-heavy, turn it back into a mixed number.
The same steps work for three fractions: 2/3 × 3/4 × 4/5 = 24/60 = 2/5.
Practice questions
Multiply the fractions
Try each one, then check your answer.
- 1/2 × 1/4
Show answer
(1 × 1)/(2 × 4) = 1/8 - 2/3 × 1/5
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(2 × 1)/(3 × 5) = 2/15 - 3/4 × 2/5
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6/20 = 3/10 - 5/6 × 3/4
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15/24 = 5/8 - 4 × 2/9
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4/1 × 2/9 = 8/9 - 6 × 3/4
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18/4 = 9/2 = 4 1/2 - 2/3 of 12
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12 ÷ 3 = 4, then 4 × 2 = 8 - 3/5 of 40
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40 ÷ 5 = 8, then 8 × 3 = 24 - 7/10 × 5/14
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Cancel 7 with 14 and 5 with 10: 1/2 × 1/2 = 1/4 - 8/9 × 3/4
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Cancel 8 with 4 and 3 with 9: 2/3 × 1/1 = 2/3 - 1 1/2 × 4
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3/2 × 4/1 = 12/2 = 6 - 2 1/3 × 3/7
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7/3 × 3/7 = 21/21 = 1 - 1 3/4 × 2 2/3
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7/4 × 8/3 = 56/12 = 14/3 = 4 2/3 - 2/3 × 3/4 × 4/5
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24/60 = 2/5 - −3/4 × 2/3
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−6/12 = −1/2 (a negative times a positive is negative) - x/2 × 4/5
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4x/10 = 2x/5
Word problems
Multiplying fractions in real life
Try each one, then check your answer.
- A recipe needs 3/4 cup of sugar. You make half the recipe. How much sugar do you need?
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1/2 × 3/4 = 3/8 cup - In a class of 30 pupils, 2/3 walk to school. How many walk?
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30 × 2/3 = 60/3 = 20 pupils - A rug is 1 1/2 m long and 2/3 m wide. What is its area?
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3/2 × 2/3 = 6/6 = 1 m² - Each bottle holds 3/4 litre. How much do 6 bottles hold?
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6 × 3/4 = 18/4 = 4 1/2 litres - 3/5 of a pizza is left, and you eat 1/3 of what is left. What fraction of the whole pizza did you eat?
Show answer
1/3 × 3/5 = 3/15 = 1/5 of the pizza
Questions people ask
What is the rule for multiplying fractions?
Multiply the numerators, multiply the denominators, then simplify. In symbols: a/b × c/d = (a × c)/(b × d).
Do you need a common denominator to multiply fractions?
No. Common denominators are only needed for adding and subtracting. Fractions with the same or different denominators are multiplied in exactly the same way.
How do you multiply a fraction by a whole number?
Write the whole number over 1 and multiply as normal. 5 × 2/3 = 5/1 × 2/3 = 10/3 = 3 1/3. Only the numerator changes.
How do you multiply mixed numbers?
Change each mixed number to an improper fraction, multiply, then change the answer back. 1 1/2 × 1 1/3 = 3/2 × 4/3 = 12/6 = 2.
What is cross multiplying fractions?
Cross multiplication is used to compare two fractions or solve proportions, not to multiply them. What helps when multiplying is cross-cancelling: dividing a top and a bottom by a common factor before you multiply.
Why does multiplying by a fraction make the answer smaller?
Multiplying by 1/2 means taking half of something, so you end up with less. Any fraction less than 1 shrinks the number; an improper fraction such as 3/2 makes it bigger.
What grade is multiplying fractions?
In the US, fraction × whole number starts in grade 4 and fraction × fraction is a grade 5 topic. In England, pupils multiply simple pairs of fractions in Year 6.
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Multiplying fractions
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What is 2/3 × 4/5?
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Related topics
Fractions
A fraction is part of a whole that has been split into equal parts. In ¾, the bottom number says how many equal parts and the top says how many you have.
Simplifying fractions
To simplify a fraction, divide the top and bottom by the same number until you cannot go any further. For example, 12/18 simplifies to 2/3 when you divide both by 6.
Improper fractions and mixed numbers
An improper fraction has a numerator bigger than its denominator, like 7/4. A mixed number shows the same amount as a whole number plus a fraction, like 1 3/4.
Dividing fractions
To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change ÷ to ×, flip the second fraction, then simplify. 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2.
Reciprocals
The reciprocal of a number is 1 divided by that number, so the two multiply to make 1. To find the reciprocal of a fraction, flip it upside down: the reciprocal of 3/4 is 4/3.
Adding and subtracting fractions
To add or subtract fractions, first give them the same denominator. Then add or subtract only the numerators and keep the denominator. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.