Multiplying a fraction by a natural number — Mathematics, 11–13
Multiplying a fraction by a natural number is adding that fraction again and again, or taking that part of a quantity.
Adding a fraction several times
Multiplying a natural number by a fraction is adding that fraction several times: 3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5. The fraction can also act as an operator on a quantity: 3/4 × 12 means “3/4 of 12”, that is, split 12 into 4 equal parts and keep 3 of them. Both readings describe the same kind of product.
Why start with one slice
A fraction like 3/4 is “3 times 1/4”. So it is easier to find the part for one slice first, 1/4 of 12 = 12 ÷ 4 = 3, and then put together the slices you need. This idea solves everything without formulas: split by the denominator and multiply by the numerator. A drawing of 12 dots in 4 groups shows it at once.
Two problems, step by step
Problem 1: what is 3/4 × 12? Step 1, the unit fraction: 1/4 × 12 = 12 ÷ 4 = 3. Step 2: 3/4 is three times 1/4, so 3/4 × 12 = 3 × 3 = 9. Problem 2: each bottle holds 2/5 of a litre; how many litres in 3 bottles? 3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5 = 1 + 1/5. They hold 1 litre and 1/5 of a litre.
Multiplying top and bottom
A reasonable mistake is to multiply the natural number by both terms: 3 × 2/5 = 6/15. But 6/15 is equivalent to 2/5: multiplying numerator and denominator by the same number does not change the value, so the result stayed the same when it should be three times bigger. The denominator tells the size of each part, and that does not change when you gather more parts. Only the numerator grows: 3 × 2 = 6 fifths.
Recipes, time and measures
It shows up in recipes and measures all the time: if one cake needs 3/4 of a cup of flour, 4 cakes need 4 × 3/4 = 12/4 = 3 cups. With time: 2 study sessions of 3/4 of an hour add up to 2 × 3/4 = 6/4 = 1 + 1/2 hours, that is, 1 hour 30 minutes. Before calculating, ask yourself: am I repeating a part, or taking a part of a quantity?
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