To multiply fractions $a/b$ and $c/d$ we simply multiply the numerators and denominators, giving $(ac)/(bd).$

This is most easily thought of as saying that multiplication by a fraction is to stretch by the numerator, and shrink by the denominator. We think of $2/3$ not as two lots of one third of something, but as an operation in which we make something twice as big, and then shrink the result to one third of its size.

Note that we are thinking of $2/3$ here as an operation to perform, not as a place to be on the number line.

In this way, multiplying two fractions has the meaning of performing the first operation, and then the second. Thus when we multiply $a/b$ and $c/d$ we expand by $a$, then shrink by $b$, then expand by $c$, and finally shrink by $d.$

But it doesn't matter what order we do these things, so that's the same as expanding by $a$ and $c$, and then shrinking by $b$ and $d$. Successive expansions and contractions are obtained by multiplying the associated integers, and hence the identity that:

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