What number should be added to each of 6, 14, 18, and 38 so that they become proportionate (i.e., 6+x : 14+x = 18+x : 38+x)?

Introduction / Context:
We want one constant x that, when added to each term, creates two equal ratios. This is a classic “bring into proportion” problem based on cross-multiplication and solving a linear equation.

Given Data / Assumptions:

• Numbers: 6, 14, 18, 38
• We need (6 + x)/(14 + x) = (18 + x)/(38 + x)
• x is the same for all four numbers.

Concept / Approach:
Set up the proportion and cross-multiply to clear denominators. Expand, simplify, and solve the resulting linear equation for x.

Step-by-Step Solution:
(6 + x)(38 + x) = (14 + x)(18 + x)Left: 6*38 + 44x + x^2 = 228 + 44x + x^2Right: 14*18 + 32x + x^2 = 252 + 32x + x^2Equate and cancel x^2: 228 + 44x = 252 + 32x ⇒ 12x = 24 ⇒ x = 2

Verification / Alternative check:
Substitute x = 2: Left ratio (8:16) = 1:2; Right ratio (20:40) = 1:2. Proportion satisfied.

Why Other Options Are Wrong:
Adding 1, 3, 4, or 5 yields ratios that do not match upon reduction.

Common Pitfalls:
Errors in expansion; forgetting to cancel x^2 on both sides; miscomputing 6*38 or 14*18.

Final Answer:
2