Combine chained ratios: If P : Q = 8 : 15 and Q : R = 3 : 2, find the compound ratio P : Q : R in simplest integers.
Correct Answer: 8 : 15 : 10
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