You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.
Oh dear. If anyone on the Internet has had more to say about dividing fractions than I have, I am unaware of who that is. (And, for the record, I would like to buy that person an adult beverage!)
Unlike the division by zero stuff from question 1, this question is better tackled with informal notions than with formalities. The formalities leave one feeling cold and empty, for they don’t answer the conceptual why. The formalities will invoke the associative property of multiplication, the definition of reciprocal, inverse and the multiplicative identity, et cetera.
The conceptual why—for many of us—lies in thinking about fractions as operators, and in thinking about a particular meaning of division.
1. A meaning of division
There are two meanings for division: partitive (or sharing) and quotative (or measuring). The partitive meaning is the most common one we think of when we do whole number division. I have 12 cookies to share equally among 3 people. How many cookies does each person get? We know the number of groups (3 in this example) and we need to find the size of each group.
When dividing by a fraction, partitive division means that we know the fractional part of a group we have, and we need to find the size of a whole group.
I can mow 4 lawns with of a tank of gas in my lawnmower is a partitive division problem because I know what of a tank can do, and I want to find what a whole tank can do. So performing the division will answer the question.
2. Fractions as operators
When I multiply by a fraction, I am making things larger (if the fraction is greater than 1), or smaller (if the fraction is less than 1, but still positive).
Scaling from (say) 5 to 4 requires multiplying 5 by . Scaling from 4 to 5 requires multiplying by . This relationship always holds—reverse the order of scaling and you need to multiply by the reciprocal.
putting it all together
Back to the lawnmower. There is some number of lawns I can mow with a full tank of gas in my lawnmower. Whatever that number is, it was scaled by to get 4 lawns. Now we need to scale back to that number (whatever it is) in order to know the number of lawns I can mow with a full tank.
So I need to scale 4 up by .
Now we have two solutions to the same problem. The first solution involved division. The second solution involved multiplication. They are both correct so they must have the same value. Therefore,
There was nothing special about the numbers chosen here, so the same argument applies to all positive values.
We have to be careful about zero. Negative numbers behave the same way as positive numbers in this case, since the associative and commutative properties of multiplication will let us isolate any values of and treat everything else as a positive number.
More on partitive fraction division here.
Please note that you do not need to invert and multiply to solve fraction division problems. You can use common denominators, then divide just the resulting numerators. You can use common numerators, then use the reciprocal of the resulting denominators. Or you can just divide across as you do when you multiply fractions. The origins of the strong preference for invert-and-multiply are unclear.