Multiplying Fractions

Start with the word rather than the symbols. Half of one third — what does that actually look like? Take a whole strip, cut it into thirds and shade one of them. Now slice every third in half across the middle. The whole strip is now six equal pieces, and the part you are holding is one of them. So half of one third is one sixth. The important idea is that the word of means multiply, so you have just worked out one half times one third equals one sixth. Notice something surprising too. The answer is smaller than both fractions you started with, because you took a part of a part. Multiplying does not always make things bigger.

Here is two thirds times three quarters drawn as a grid. Cutting the square into three columns and shading two gives you two thirds. Cutting it into four rows and shading three gives three quarters. The part where the two shadings overlap is the answer. The whole grid holds three times four, which is twelve cells, and the overlap holds two times three, which is six. So two thirds times three quarters is six twelfths, and that simplifies to one half. The grid has just shown you the rule. Multiply the tops together, multiply the bottoms together. Three fifths times two sevenths is six thirty-fifths. And notice you never needed a common denominator — that is only for adding and subtracting.

Multiplying three eighths by four ninths the long way gives twelve over seventy-two, and then you still have to simplify that, which means hunting for the highest common factor of two awkward numbers. There is a much kinder route. Before you multiply anything, look for a top and a bottom that share a factor and divide them both. The three and the nine both divide by three, leaving one and three. The four and the eight both divide by four, leaving one and two. Now the question reads one half times one third, which is one sixth, already in simplest form. You may cancel any numerator against any denominator, including diagonally across the multiplication sign.

Finding a fraction of a whole number uses exactly the same idea. Three fifths of twenty counters. Share the twenty into five equal groups and each group holds four, then take three of those groups, which is twelve. In words the method is divide by the bottom, then multiply by the top. Two thirds of twelve: twelve divided by three is four, times two is eight. You can also write it purely as a multiplication by remembering that every whole number is secretly a fraction over one. Twenty is twenty over one, so three fifths times twenty over one, and the five cancels into the twenty leaving four, giving three times four, which is twelve.

A mixed number like two and a quarter is not safe to multiply as it stands, so change it into an improper fraction first. Multiply the whole part by the bottom and add the top: two times four is eight, plus one is nine, so two and a quarter is nine quarters. Now try one and a half times two thirds. One and a half becomes three halves, so you have three halves times two thirds. Cancel the twos and cancel the threes and everything collapses to one. Or take two and a quarter times two. That is nine quarters times two over one, the two cancels into the four, and you get nine halves, which is four and a half. Convert the answer back at the end.