If 0.764 × P = 1.236 × Q, determine the exact decimal value of (Q − P) / (Q + P). Do not compute P or Q separately; use ratio manipulation.

Introduction / Context:
This problem mirrors a common ratio trick: you are given a linear relation between P and Q and asked for (Q − P) / (Q + P). The key is to express one variable in terms of the other and reduce the expression to a function of a single ratio.

Given Data / Assumptions:

• Equation: 0.764 * P = 1.236 * Q.
• Expression needed: (Q − P) / (Q + P).
• All values are real and finite.

Concept / Approach:
From 0.764P = 1.236Q, obtain the ratio k = P / Q = 1.236 / 0.764. Then substitute P = kQ into (Q − P) / (Q + P) to get (Q − kQ) / (Q + kQ) = (1 − k) / (1 + k). This avoids finding numeric P and Q.

Step-by-Step Solution:

Verification / Alternative check:
Pick convenient proportional values obeying 0.764P = 1.236Q, for example let Q = 0.764 and P = 1.236. Then (Q − P) / (Q + P) = (0.764 − 1.236) / (0.764 + 1.236) = (−0.472) / 2.000 = −0.236.

Why Other Options Are Wrong:
0.236 and 0.472 are magnitudes seen in the arithmetic but the correct signed value is negative. 0.764 and 2 are unrelated to the final ratio form.

Common Pitfalls:
Inverting the ratio (using Q/P) or dropping the negative sign when computing (Q − P). Pay attention to which variable is subtracted.

Final Answer:
-0.236