What should be subtracted from 10, 12, 13, 16 so that the remaining numbers may be proportional?

Correct — C. Call the number subtracted x.

Proportional means product of extremes = product of means:
(10 − x)(16 − x) = (12 − x)(13 − x)

Expand: 160 − 26x + x² = 156 − 25x + x²
Cancel x²: 160 − 26x = 156 − 25x
Collect: 160 − 156 = 26x − 25x, so x = 4

Check: 6, 8, 9, 12 give 6 × 12 = 72 = 8 × 9 → option (c)

Why the others are wrong

• (a)
  3 — Subtracting 3 gives 7, 9, 10, 13: extremes 7 × 13 = 91, means 9 × 10 = 90. One apart, so 7 : 9 and 10 : 13 are not equal.
• (b)
  9 — Subtracting 9 gives 1, 3, 4, 7: extremes 1 × 7 = 7, means 3 × 4 = 12. The ratios 1 : 3 and 4 : 7 are far apart.
• (d)
  12 — Subtracting 12 gives −2, 0, 1, 4: extremes −2 × 4 = −8, means 0 × 1 = 0. The products differ, and −2 : 0 is not even a defined ratio.

Four numbers a, b, c, d are in proportion when a : b = c : d, which is the same as ad = bc: the product of the extremes (1st and 4th) equals the product of the means (2nd and 3rd).

When the same x comes off all four, the x² terms cancel and a one-step linear equation is left. Testing the options is just as fast: subtract each and compare the two products.

Key facts

• a, b, c, d are in proportion when ad = bc.
• If subtracting x from a, b, c, d makes them proportional, x = (ad − bc) ⁄ ((a + d) − (b + c)). Here (160 − 156) ⁄ (26 − 25) = 4.
• 6, 8, 9, 12 are proportional: 6 : 8 = 9 : 12 = 3 : 4.

Common traps

• Pairing the wrong products: the extremes are the 1st and 4th numbers (10, 16), the means the 2nd and 3rd (12, 13).
• Subtracting x from only some of the numbers: all four become 10 − x, 12 − x, 13 − x and 16 − x.

No directly related past PYQ was found.