Find the quantity x that must be added to both terms of the ratio a : b so that the new ratio becomes c : d. Express x in terms of a, b, c, and d.

Introduction / Context:
We want to adjust a ratio by adding the same amount x to both terms so that it changes to a specified ratio c : d. This requires algebraic manipulation of proportions.

Given Data / Assumptions:
Desired: (a + x) : (b + x) = c : d with real values and c ≠ d to avoid division by zero in the final expression.

Concept / Approach:
Translate the ratio to a fraction equality and cross multiply. Solve the linear equation for x.

Step-by-Step Solution:
(a + x)/(b + x) = c/d. Cross multiply: d(a + x) = c(b + x). da + dx = cb + cx. Group x terms: dx − cx = cb − da. x(d − c) = cb − da = −(ad − bc). x = (cb − da)/(d − c) = (ad − bc)/(c − d).

Verification / Alternative check:
Substitute x = (ad − bc)/(c − d) back into (a + x)/(b + x) and simplify; the result reduces to c/d.

Why Other Options Are Wrong:
The other formulas do not satisfy the derived linear equation for x; they result from misgrouping or mis-cross-multiplication.

Common Pitfalls:
Sign errors when moving terms or inverting (c − d) to (d − c). Both forms are equivalent up to a sign flip; ensure consistency with the numerator.

Final Answer:
(ad − bc) / (c − d)