If P : Q : R = 2 : 3 : 5, then what is the ratio (P + Q) : (Q + R) : (R + P)?

Introduction / Context:
This question checks the understanding of manipulating ratios. We are given a ratio between three quantities P, Q and R, and asked to find the ratio between different sums of these quantities. This is a typical ratio transformation question, where each term is a simple linear combination of the original variables. It tests comfort with expressing variables in terms of a common multiplier and then recomputing ratios.

Given Data / Assumptions:
• P : Q : R = 2 : 3 : 5. • All three quantities P, Q and R are positive. • We need the ratio (P + Q) : (Q + R) : (R + P).

Concept / Approach:
When a ratio of three quantities is given, we usually represent each quantity in terms of a single constant multiplier. Here, P = 2k, Q = 3k and R = 5k for some positive k. Then we compute each required combination, such as P + Q, in terms of k, and finally form the ratio of these combinations. Because k is a common factor, it will cancel out, leaving a ratio in simple integers.

Step-by-Step Solution:
Step 1: Represent P, Q and R using a common multiplier k: P = 2k, Q = 3k, R = 5k. Step 2: Compute P + Q = 2k + 3k = 5k. Step 3: Compute Q + R = 3k + 5k = 8k. Step 4: Compute R + P = 5k + 2k = 7k. Step 5: Now form the required ratio: (P + Q) : (Q + R) : (R + P) = 5k : 8k : 7k. Step 6: Cancel the common factor k to get 5 : 8 : 7.

Verification / Alternative Check:
To verify, we can assign an actual value to k, such as k = 1. Then P = 2, Q = 3 and R = 5. Now P + Q = 5, Q + R = 8 and R + P = 7. Their ratio is clearly 5 : 8 : 7, which matches the derived result. Any other positive value of k would give the same ratio because it simply scales all three sums by the same factor.

Why Other Options Are Wrong:
• 2 : 3 : 5 is just the original ratio P : Q : R and not the ratio of their sums. • 5 : 8 : 10 would correspond to P + Q, Q + R and Q + R again if miscalculated, but not to P + Q, Q + R and R + P. • 4 : 9 : 25 is unrelated to the linear combinations of 2, 3 and 5.

Common Pitfalls:
A frequent error is to plug numbers directly without using a common variable, leading to inconsistent results if one is not careful. Another mistake is mixing up which sums are required; for example, some may compute (P + Q) : (Q + R) : (P + Q + R). Writing each target expression clearly before substituting helps avoid confusion.

Final Answer:
The required ratio (P + Q) : (Q + R) : (R + P) is 5 : 8 : 7.