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3.7 Pre-cancelling when Multiplying Fractions

In the fraction product
21 15
· ,
5 14
we could simply follow the rule that the product of fractions is the product of the numerators over the
product of the denominators,
21 · 15
,
5 · 14
calculate the products in the numerator and denominator and, finally, reduce the resulting fraction to
lowest terms by cancelling out the GCF. But there’s a short-cut: Since multiplication is commutative,
we can reverse the order of multiplication in the numerator, obtaining

15 · 21 15 21
= · .
5 · 14 5 14
The fractions on the right-hand side are easily reduced to lowest terms by cancelling the obvious GCFs
(5 and 7, respectively)
✚❃
✚❃
✚ 15 3 ✚ 21 3
· , 2
1
✚❃
✚ 5 ✚❃ ✚ 14
leaving a simpler product
3 3 9
· = ,
1 2 2
9
which has the further advantage that the solution, , is already in lowest terms. The trick of cancelling
2
before multiplying – pre-cancellation – saves us from bigger numbers,
21 15 315
· = ,
5 14 70
and the extra work of finding the GCF{315, 70} =35 for cancellation:

✟ ✟✯ 9
✟ 315 = 9 (cancelling 35),
✟✯ 2 2
✟ 70
The general rule is this:

In a product of fractions, a factor which is common to one
of the numerators and one of the denominators can be can-
celled before multiplying. The numerator and denominator
need not belong to the same fraction.

Notice that pre-cancellation works with any number of factors.

Example 98. Find the product
3 8 10
· · .
4 5 9

Page 89