If $P : Q = 3 : 4$ and $Q : R = 4 : 7$, then find the value of $(P + Q) : (Q + R) : (R + P)$.

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**Solution:**

1. Given ratios are $P : Q = 3 : 4$ and $Q : R = 4 : 7$.

2. Since $Q$ is common in both ratios with the same value 4, combine to get:
   $$P : Q : R = 3 : 4 : 7$$

3. Let $P = 3k$, $Q = 4k$, and $R = 7k$ for some constant $k$.

4. Calculate the sums:
   - $P + Q = 3k + 4k = 7k$
   - $Q + R = 4k + 7k = 11k$
   - $R + P = 7k + 3k = 10k$

5. Form the ratio:
   $$(P + Q) : (Q + R) : (R + P) = 7k : 11k : 10k$$

6. Cancel the common factor $k$:
   $$7 : 11 : 10$$

**Answer:** $7 : 11 : 10$

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