Dividing by a fraction

A jug holds 3 litres of water. It's poured out into little cups, and each cup holds one quarter of a litre. How many cups can we fill? Let's draw the water. Here are the 3 litres. 1, 2, 3. 3 litres in all. Now, each cup holds one quarter of a litre. One quarter means 1 out of 4 equal parts. So we cut every litre into 4. There's one cup — one quarter of a litre. Now count the cups that fit along the bar. 1, 2, 3, 4, and on all the way to 12. Each litre holds 4 cups, so 3 litres hold 3 times 4. That's 12 cups. We can write that count as a division. 3 divided by one quarter. To divide by a fraction, we take its reciprocal, which means we turn that fraction upside down. So one quarter turns into 4 over 1, and 3 divided by one quarter becomes 3 times 4. And that's 12. The cups say the same thing. There are 12 of them along the bar. Look, it's more than 3, not less. Lots of small cups fill a big jug. So 12 cups can be filled.

Your turn. Here's a new question. A different jug holds 2 litres, and this time the cups hold one half of a litre each. How many cups can we fill? Try it first. Pause if you like. One half means 1 out of 2 equal parts. So we cut every litre into 2. Count them. 1, 2, 3, 4. Each litre holds 2 cups, so 2 litres hold 2 times 2. That's 4 cups. As a division, that's 2 divided by one half. Take the reciprocal. Turn the one half upside down, and it is 2 times 2. And that's 4. Dividing by one half is the same as multiplying by 2. So 4 cups can be filled. So whenever you divide by a fraction, don't shrink it. Count how many of that fraction fit. 4 one-half-litre cups fit in 2 litres. That's what 2 divided by one half means.