Combine chained ratios: If P : Q = 8 : 15 and Q : R = 3 : 2, find the compound ratio P : Q : R in simplest integers.

Introduction / Context:
Chained ratios require aligning the common term. Here Q appears in both given ratios, so we scale them to the same Q before combining into a three-term ratio P : Q : R.

Given Data / Assumptions:

• P : Q = 8 : 15.
• Q : R = 3 : 2.
• All terms represent positive quantities.

Concept / Approach:
Make Q equal in both ratios. Since Q is 15 in the first and 3 in the second, multiply the second ratio by 5 to get Q = 15 in both. Then read off P, Q, and R directly.

Step-by-Step Solution:
Scale Q : R = 3 : 2 by 5 ⇒ Q : R = 15 : 10.Now P : Q = 8 : 15 and Q : R = 15 : 10.Therefore, P : Q : R = 8 : 15 : 10.

Verification / Alternative check:
Let Q = 15 units. Then P = 8 units (from the first ratio) and R = 10 units (from the scaled second), confirming the compound ratio.

Why Other Options Are Wrong:

• 8 : 15 : 7 and 7 : 15 : 8 invent an R that does not follow Q : R = 3 : 2.
• 10 : 15 : 8 swaps positions incorrectly.

Common Pitfalls:

• Adding ratios instead of aligning the common term.
• Forgetting to scale the second ratio properly.

Final Answer:
8 : 15 : 10