Why is it that $\frac ab\times\frac1c=\frac a{bc}$? [duplicate]
Solution 1:
You can think of multiplication as meaning "of". So what is $2/5$ of $3/7$ (for example)?
Draw a picture of a cake (a rectangular cake) sliced into 7 equal vertical slices, with $3$ of those slices having red frosting. That's $3/7$ of the cake.
Take that 3/7 of the cake and slice it horizontally into 5 equal pieces, and pour sprinkles on 2 of those 5 pieces. (When you're doing the horizontal slicing, slice the entire cake horizontally while you're at it.)
The portion of the cake with sprinkles is 2/5 of 3/7. But if you draw the picture, you see that the cake has been chopped into 35 equal pieces (5 groups of 7), and 6 of those 35 pieces have sprinkles. So, $$ \frac{2}{5} \text{ of } \frac{3}{7} = \frac{2 \times 3}{5 \times 7}. $$
Solution 2:
There are three steps to this process
- Define what a fraction actually is
- Define what multiplying two numbers actually does
- Prove that multiplying fractions is done by multiplying numerator with numerator and denominator by denominator
Steps 1 and 2 can be done many different ways, and for each combination, step 3 will be done differently.
Solution 3:
Great question! The short answer to this: it works because we defined it like this. (I assume we are talking about multiplication of rational numbers)
We are worried, however, whether the operation is well-defined. It means that the result $$\frac{a}{b} \cdot \frac{x}{y} = \frac{ax}{by} $$ must not depend on the choice of fractions. It can't be that this equality holds for some fractions, but not for some other fractions. That would make the operation ill-defined.
The process of verification is quite an involved one, however, especially for multiplication.
On a quick search I did find this which covers all one needs.
To give a slightly different spin on this problem. Intuition paves the way for how we want to define certain operations. Other answers give an intuitive explanation why multiplying two fractions produces a certain fraction. We used these intuitions to define how multiplication of two fractions behaves. But to be absolutely sure we didn't make a mistake, we must also verify the operation is well-defined and that is beyond reach for intuition.
This idea of well-definedness is very important in mathematics not just as a failsafe for addition and multiplication of (rational) numbers to be bulletproof.