What number should be subtracted from each of 50, 61, 92, 117 so that the…

In a proportion a : b :: c : d, the product of the extreme terms equals the product of the middle (mean) terms: a × d = b × c. This extremes-means rule lets four quantities be tested for, or adjusted into, proportion without evaluating each ratio separately.

1. Let x be the number subtracted from each of 50, 61, 92 and 117, giving the four terms (50 − x), (61 − x), (92 − x) and (117 − x).

2. For these terms to be in proportion in the given order, (50 − x) : (61 − x) :: (92 − x) : (117 − x), so by the extremes-means rule, (50 − x)(117 − x) = (61 − x)(92 − x).

3. Expanding both products: 5850 − 167x + x2 = 5612 − 153x + x2.

4. The x2 term cancels from both sides, leaving 5850 − 167x = 5612 − 153x.

5. Collecting the x-terms on one side and the constants on the other: 5850 − 5612 = 167x − 153x, i.e. 238 = 14x.

6. Dividing both sides by 14: x = 238 ÷ 14 = 17.

Substituting x = 17 back into the four terms gives 33, 44, 75 and 100. Checking the extremes-means rule: 33 × 100 = 3300 and 44 × 75 = 3300 — the two products are equal, confirming the terms are in proportion.

So the number that must be subtracted from each of 50, 61, 92 and 117 is 17.