Algebra with a given ratio: If P : Q = 7 : 5, find the value of (5P − 2Q) : (3P + 2Q).

Introduction / Context:
When the ratio of two variables is known, you can represent them with a common multiplier and evaluate composite linear combinations cleanly. This is a standard technique for simplifying expressions involving ratios.

Given Data / Assumptions:

• P : Q = 7 : 5.
• Compute (5P − 2Q) : (3P + 2Q).

Concept / Approach:
Let P = 7k and Q = 5k for some k > 0. Substitute and simplify each linear combination to obtain a simple fractional ratio, which is independent of k.

Step-by-Step Solution:
P = 7k, Q = 5k.5P − 2Q = 5(7k) − 2(5k) = 35k − 10k = 25k.3P + 2Q = 3(7k) + 2(5k) = 21k + 10k = 31k.Ratio = 25k : 31k = 25/31.

Verification / Alternative check:
Pick k = 1 to verify numerically: P = 7, Q = 5 ⇒ (35 − 10) : (21 + 10) = 25 : 31, confirming the result.

Why Other Options Are Wrong:

• 5/4, 6/5, 31/42 emerge from incorrect arithmetic on coefficients.

Common Pitfalls:

• Losing the common factor k when simplifying, which can lead to wrong numerators/denominators.

Final Answer:
25/31