# Multiplying and Dividing Fractions

Multiplying fractions is easier than Adding and subtracting them, but before we look at the method for solving them, it's worth thinking a little about what it actually means to multiply a fraction by a fraction. A half times a half is best understood as meaning a half of a half. You might not normally think of the words "times" and "of" meaning the same thing, but it works with non-fractions too, so long as you add in the word "lots": four times five = four lots of five. Okay, let's now look at the method with the sum 2/3 times 3/4 (which is more easily understood as meaning 2/3 of 3/4):-

__ _  2  3 × __ _  3  4 =
__ _  6 12 = __ _  1  2

Click the button and read the comments here for each step towards solving the sum.

Let's now do another one, but this time a division rather than a multiplication. Whenever you divide a fraction by a fraction, you can simply convert the whole sum into a multiplication by turning the second fraction upside down. Why does this work? Well, multiplying a number by 2 is the same as dividing it by a half, and all numbers that don't look like fractions can be turned into fractions without changing their value at all just by putting them over 1, so 2 = 2/1, and 2/1 is 1/2 upside down. Let's see the process in action then:-

__ _  1 10 ÷ __ _  1  5 =
__ _  1 10 × __ _  5  1 =
__ _  5 10 = __ _  1  2

Click the button and read the comments here for each step towards solving the sum.

Let's now do another one where we're dividing by a smaller fraction instead of a bigger one, It makes no difference to the method, but I want you to get a feel for the rightness of the answers as well as learning the mechanical method for solving sums:-

__ _  4  5 ÷ __ _  3  8 =
__ _  4  5 × __ _  8  3 =
__ _ 32 15 = 2 __ _  2 15

Click the button and read the comments here for each step towards solving the sum.

In the sum we've just looked at there was a top-heavy fraction (32/15): that's a fraction where the top number is bigger than the bottom one. Top-heavy fractions aren't good as final answers, so you need to simplify them by subtracting the bottom number away from the top number as many times as you can while keeping count of how many times you've done it, and that count will be the number that goes in front of whatever's left of the fraction that can't be removed in this way. You will get lots of practice at doing this later, so don't worry about it for now.

If you want to add 2 1/4 + 3 1/2, you can simply add the 2 and 3 together and then add the fractions together to get 5 3/4, so that's very easy, but multiplying 2 1/4 × 3 1/2 is a very different story because the 2 has to be multiplied by both the 3 and the 1/2, and then the 1/4 has to be multiplied by both the 3 and the 1/2 as well. The easiest way to make sure this happens is simply to turn the numbers into top-heavy fractions. Here's an example for you to click your way through repeatedly until you understand the process:-

2 __ _  2  7 × 3 __ _  3  8 =
__ _ 16  7 × __ _ 27  8 =
__ __ 432 56 = 7 __ _  5  7

Click the button and read the comments here for each step towards solving the sum.

You now need some practice at solving more of these, so ... (practice program yet to be written - come back later).