Dividing Fractions

Before any rules, get the picture. Twelve divided by three asks how many threes fit inside twelve, and the answer is four. Dividing fractions asks the very same question. One half divided by one quarter asks how many quarters fit inside one half. Shade a half of a strip, then cut that whole strip into quarters, and you can see that exactly two quarters cover the shaded part. So one half divided by one quarter is two. The same reasoning gives three quarters divided by one eighth: six eighths cover three quarters, so the answer is six. Notice that dividing by a fraction smaller than one makes the answer bigger, because small pieces fit in many times over.

The next idea you need is the reciprocal, which is just a fraction turned upside down. The reciprocal of three quarters is four thirds. The reciprocal of two sevenths is seven halves. There is a neat test that tells you whether you have it right: a number multiplied by its reciprocal always gives exactly one. Three quarters times four thirds is twelve over twelve, which is one. If your answer is not one, you have flipped something wrongly. Whole numbers have reciprocals too, once you remember that every whole number sits over one. Five is five over one, so its reciprocal is one fifth. Nine flips to one ninth, and one eighth flips all the way up to eight.

Drawing pictures gets slow, so here is the method: keep, change, flip. Keep the first fraction exactly as it is. Change the divide sign into a times sign. Flip the second fraction upside down. That is it. Two thirds divided by four fifths becomes two thirds times five quarters, which is ten twelfths, and that simplifies to five sixths. You could also cancel the two into the four before multiplying and get five sixths straight away. Flipping works because asking how many quarters fit in a half is the same as asking what a half times four is. The one thing to be careful about is that only the second fraction flips. Leave the first one alone.

Whole numbers turn up on both sides of the divide sign, and the trick is the same each time: write the whole number over one. Three quarters divided by three becomes three quarters times one third, which is three twelfths, or one quarter. That makes sense, because sharing three quarters of a pizza between three people gives each person a quarter. Going the other way, six divided by one half becomes six over one times two over one, which is twelve. That also makes sense: six pizzas each cut in half give twelve pieces. Use those two pictures as a size check. Dividing by a whole number makes the answer smaller, while dividing by a fraction under one makes it bigger.

Real questions usually arrive as words. A ribbon two and a half metres long is cut into half-metre pieces — how many pieces? Convert the mixed number first, so two and a half becomes five halves. Then keep, change, flip: five halves times two over one is ten over two, which is five. The picture agrees: five half-metre pieces. The same routine handles anything. One and a half divided by three quarters becomes three halves times four thirds, which is two. Five sixths divided by two thirds becomes five sixths times three halves, which is fifteen twelfths, or one and a quarter. Look out for the phrases that mean divide: cut into pieces of, shared between, and how many fit.