Given a : b = c : d, evaluate (m a + n c) / (m b + n d). The expression simplifies to which ratio?

Introduction / Context:
This property of proportional quantities states that if two ratios are equal, then any linear combination with the same coefficients in numerator and denominator preserves the common ratio. It is a fundamental result in ratio manipulation.

Given Data / Assumptions:

• a : b = c : d.
• m and n are constants.
• We evaluate (m a + n c) / (m b + n d).

Concept / Approach:
If a/b = c/d = k, then a = kb and c = kd. Substitute into the expression to check simplification toward k (i.e., a : b).

Step-by-Step Solution:
Let a = kb and c = kd. Then (m a + n c) / (m b + n d) = (m kb + n kd) / (m b + n d). Factor k in numerator: = k (m b + n d) / (m b + n d) = k. Since a/b = k, the ratio equals a : b.

Verification / Alternative check:
Choose sample numbers satisfying a/b = c/d (e.g., 2/3 and 4/6). Any positive m, n will yield the same ratio value, confirming the identity.

Why Other Options Are Wrong:
m : n, na : mb, md : nc do not generally hold for arbitrary m, n; only a : b is invariant.

Common Pitfalls:
Treating m and n as multiplicative scalars of the whole ratio rather than components of a linear combination. The invariance depends on a/b = c/d.

Final Answer:
a : b