🍫 Multiplying Fractions: A Part of a Part

Hello! If we take a part of a part of a chocolate bar, how much chocolate do we have? Today we will discover how to multiply fractions with a bar of chocolate.

Our bar has 4 equal columns and 3 equal rows: 12 pieces in all. Let’s take 3 of the columns. We have 3/4 of the chocolate.

Now let’s take 2/3 of the part we took. The part has 3 rows, and we colour 2 of them blue. The place where the colours overlap is a part of a part.

Let’s count the overlap: 6 pieces out of 12. So 3/4 × 2/3 = 6/12. Here is the shortcut: multiply the numerators, 3 × 2 = 6. Multiply the denominators, 4 × 3 = 12.

6/12 can be made even simpler. Let’s divide the numerator and the denominator by 6: 6/12 = 1/2. Yes, the part we coloured is exactly half of the chocolate!

An easier way: simplify first, then multiply! Divide the 3 on top and the 3 on the bottom by 3, and both become 1. Then divide the 4 and the 2 by 2: they become 2 and 1. We are left with 1/2 × 1/1 = 1/2. The same answer, with less work!

Multiplying by a whole number is repeated addition. 3 × 2/5 means adding 2/5 three times. Make three hops of 2/5 on the number line: 2/5 + 2/5 + 2/5 = 6/5, which is 1 and 1/5.

We often think multiplying always makes things bigger, but not here! If we take a part of a fraction, we have less. 1/2 × 3/4 = 3/8, which is smaller than both 1/2 and 3/4. Multiply by a fraction less than 1 and the answer gets smaller.

We use this at the market too! Cheese costs $8 per kilogram. How much is 3/4 kg of cheese? Cut 1 kg into 4 pieces: each piece costs $2. 3 pieces cost $6: 3/4 × 8 = 6. Since 3/4 = 0.75, you can also work out 0.75 × 8 = 6.

Summary: when we multiply fractions, numerators multiply with numerators and denominators with denominators. Simplifying first makes the work easier. Multiplying by a whole number is repeated addition. Multiply by a fraction less than 1 and the answer gets smaller. Now try it in Market Stall!