What number should be subtracted from each of 54, 71, 75, and 99 so that the resulting numbers are in proportion?

Introduction / Context:
Numbers a, b, c, d are in proportion if a : b = c : d. Here, we seek x such that (54 − x) : (71 − x) = (75 − x) : (99 − x). This leads to a simple cross-multiplication and linear equation in x.

Given Data / Assumptions:

• Numbers: 54, 71, 75, 99
• Subtract the same x from each.
• Condition: (54 − x)/(71 − x) = (75 − x)/(99 − x)

Concept / Approach:
Use cross-multiplication to eliminate denominators, expand, and solve for x. Quadratic terms cancel, yielding a first-degree equation.

Step-by-Step Solution:
(54 − x)(99 − x) = (71 − x)(75 − x)54*99 − (54 + 99)x + x^2 = 71*75 − (71 + 75)x + x^25346 − 153x = 5325 − 146x5346 − 5325 = 153x − 146x ⇒ 21 = 7x ⇒ x = 3

Verification / Alternative check:
Compute pairs after subtracting 3: 51:68 and 72:96. Reduce: 51/68 = 3/4, 72/96 = 3/4. Proportion holds.

Why Other Options Are Wrong:
1, 2, 4, or 6 do not produce equal ratios for the two pairs when checked.

Common Pitfalls:
Arithmetic slips in products 54*99 and 71*75; forgetting that x^2 cancels on both sides.

Final Answer:
3