Lesson 9 of 10 · Grades 3 to 5
Multiply a fraction by a whole number
Multiplying a fraction by a whole number is repeated addition in disguise. Three lots of 2/5 is 2/5 + 2/5 + 2/5. Once you see the pattern, you can multiply the numerator by the whole number and keep the denominator.
Groups of a fraction
Multiplying sounds like a big topic, but multiplying a fraction by a whole number is really just repeated addition. Three lots of 2/5 means 2/5 + 2/5 + 2/5. Add the numerators: 2 + 2 + 2 = 6, and keep the denominator: the answer is 6/5. You have five fifths in a whole, so 6/5 is 1 1/5.
Reading the question out loud helps. "3 x 2/5" can be said as "three groups of two fifths". The whole number counts the groups, and the fraction is the size of each group.
The fast rule
Repeated addition works every time, but there is a quicker way. Because each group contributes the same numerator, you can multiply the numerator by the whole number and keep the denominator. For 4 x 3/8, multiply 4 x 3 = 12 and keep the 8. The result is 12/8. The denominator stays because the size of each part never changes; you are just counting more of them.
Finish by simplifying and, if needed, writing the answer as a mixed number. 12/8 simplifies to 3/2 by dividing top and bottom by 4, and 3/2 is 1 1/2. So 4 x 3/8 = 1 1/2. An answer bigger than one whole is perfectly normal when you multiply by a whole number bigger than one.
Two ways to read the same question
The phrase "3 x 2/5" and the phrase "2/5 of 15" look different, but they belong to the same family. Multiplying by a whole number makes equal groups: 3 groups of 2/5. Finding a fraction of a number splits a number into equal groups: 2/5 of 15 means split 15 into 5 equal groups and take 2 of them. Both use the numerator as a count and the denominator as the size or number of shares. Keep the two pictures in mind and word problems become easier to sort out.
Worked example
Five friends each eat 3/4 of a sandwich. How many sandwiches do they eat altogether? Each friend eats three quarters, so the question is 5 x 3/4. Multiply the numerator: 5 x 3 = 15. Keep the denominator: 15/4. Now write it as a mixed number. 15 divided by 4 is 3 remainder 3, so 15/4 = 3 3/4. The friends eat three and three quarters of a sandwich in total.
Watch out for this
Do not multiply the denominator by the whole number. For 5 x 3/4, the denominator stays 4; turning it into 20 would make each part tiny and give a wrong, much smaller answer. Also take care to simplify at the end, and write amounts bigger than one as a mixed number when that reads more clearly.
Remember
- Multiplying a fraction by a whole number is repeated addition.
- Multiply the numerator by the whole number.
- Keep the denominator; the part size does not change.
- Simplify, and use a mixed number when the answer is more than one.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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2 x 1/4 = ?
Show answer
2/4, which simplifies to 1/2.
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3 x 2/7 = ?
Show answer
6/7.
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4 x 3/5 = ?
Show answer
12/5, which is 2 2/5.
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5 x 2/9 = ?
Show answer
10/9, which is 1 1/9.
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Six jugs each hold 1/3 of a litre of water. How much water is that altogether?
Show answer
6/3, which is 2 litres.
Puzzle · Number pattern
Puzzle time
The cards follow a pattern: card 1 is 2/5, card 2 is 4/5, card 3 is 6/5 and card 4 is 8/5. Which card is the first one worth more than 2?
Show a hint
Card 3 is 3 × 2/5. How many fifths make 2 wholes?
Show the answer
Answer: Card 6. Two wholes are 10/5. Card 5 is 5 × 2/5 = 10/5, which is exactly 2, not more than 2. Card 6 is 6 × 2/5 = 12/5 = 2 2/5. It is the first card worth more than 2.