Multiplying fractions
The idea
Taking a fraction of something is what multiplying by a fraction means. Finding 3/4 of 20 cupcakes is a two-move job you already own: split 20 into 4 equal groups (that is division), then take 3 of those groups (that is multiplication). The same thinking handles a fraction of a fraction: half of a half of a pancake is a quarter of it, which you can see by folding paper in half twice.
Here is the surprise worth sitting with: multiplying does not always make things bigger. Years of whole-number practice teach that times means more, but 3/4 of 20 is 15 — smaller than 20. That is not a trick; taking three quarters of a tray means leaving some behind, so the answer must shrink. Whenever you multiply by a fraction less than 1, expect a smaller result, and use that expectation to catch errors: if you compute 3/4 of 20 and get 26, something went wrong.
Worked example
A bakery tray holds 20 cupcakes. The baker puts sprinkles on 3/4 of them. How many cupcakes get sprinkles?
- Read 3/4 of 20 as instructions: split the 20 cupcakes into 4 equal groups, then take 3 of those groups.
- One fourth of 20 is 20 ÷ 4 = 5 cupcakes in each group.
- Three of those groups make 3 × 5 = 15 cupcakes with sprinkles.
- Check using the leftover group: the plain cupcakes are the remaining fourth, 5 of them, and 15 + 5 = 20, so every cupcake is accounted for.
- Notice that 15 is smaller than 20. Multiplying by a fraction less than 1 takes a part of the amount, so the result shrinks — exactly as it should.
Answer. 15 cupcakes get sprinkles, and the other 5 stay plain.
Check your understanding
- Why does multiplying a number by a fraction smaller than 1 give a smaller answer?
- How would you find 2/3 of 18 using the divide-then-multiply idea?
- What everyday situations ask you to take a fraction of an amount rather than the whole thing?
- How is finding half of a half like folding paper twice, and what part of the whole do you end up holding?