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Example 190. Decide if the given proportions are true or false, using the cross-product property. (a)
3 2 4 6
= ; (b) = .
11 7 10 15
Solution. (a) The cross-products are 3 · 7= 21 and 2 · 11 = 22. They are not equal, so the proportion
is false. (b) The cross-products are 4 · 15 and 10 · 6, both equal to 60. So the proportion is true.

5.2.2 Solving a proportion

If one of the four terms of a proportion is missing or unknown, it can be found using the cross-product
property. This procedure is called solving the proportion. In the proportion below, x represents an
unknown term (any other letter would do).
x 34
= .
3 51
There is a unique x which makes the proportion true, namely, the one which makes the cross-products,
51x and 34(3), equal. The equation
51x =34(3) =102
can be divided by 51 (the number which multiplies x)on both sides, giving
51x 102
= .
51 51
Cancellation yields
✟ ✟✯
2
✚❃
✚ 51x 1 = ✟ 102 =2.
✚❃
✚❃
✚ 51 1 ✚ 51 1
It follows that
x =2
which is the solution of the proportion.
It doesn’t matter which of the four terms is missing; the proportion can always be solved by a similar
procedure.

Example 191. Solve the proportion
9 36
=
100 y
for the unknown term y.

Solution. The cross-products must be equal, so

9y =3600.
Dividing both sides of the equation by 9 (the number which multiplies y), and cancelling, yields

✘ ✘✿ 400
✚❃
✚ 9 y 1 = ✘ 3600
✚ 9 ✚❃ 1 ✚ 9 ✚❃ 1
y =400.
The unknown term is 400. We check our solution by verifying that the cross-products

9(400) = 36(100) = 3600
are equal.

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