Why Does Invert and Multiply Work?

In our class this week as we looked at the Grade 7 and 8 curriculum, we finished by looking at dividing by fractions, and more specifically at the question, Why does invert and multiply work?

Why does this work?

Well, first in order to see why it works I decided to look first what it means to divide regular numbers. Let's look at the example 6 ÷ 2. We know that this is equal to three, but let's look at how we can reason this out. One way to look at 6 ÷ 2 is to ask how many times does 2 go into 6. From this we can see 3 sets of 2 would equal 6 so the answer is 3. This is similar to asking what number multiplied by 2 gives us 6.

Now let's go back to the example in the picture. If we rephrase this we can ask what multiplied by 1/6 is equal to 1/2. This is equivalent to the equation (1/6)x = 1/2 or x/6 = 1/2. Then we know that to solve for x we just multiply 1/2 by 6. However, in doing this what we have actually done is just invert and multiply.

Another way to look at this is to look at 1/2 ÷ 1/6 as if we are multiplying. In multiplying fractions we simply multiply across the top and bottom, so let's do the same for division. So in this case what we will have (1÷1)/(2÷6) = 1/(2/6). Now to simplify this we can multiply it by 6/6 and then we will have, 6/6*(2/6) = 6 / 2 = 3. So in fact we can see that invert and multiply also works as it is just a shortcut way of doing the division and then simplifying the answer.