P, Q and R are three batsmen. The ratio of runs scored by them in a certain…

Concept: when two ratios share a common term — here Q appears in both P : Q and Q : R — they can be merged into a single three-term ratio by scaling each ratio so that the shared term carries the same value in both.

Once one chained ratio a : b : c exists, a total T is split into (a + b + c) equal parts, one part is worth T ÷ (a + b + c), and any member's share is its own part-count multiplied by that part value.

Applying this to the match:

1. Write the two given ratios: P : Q = 5 : 4 and Q : R = 1 : 2. The shared term Q is 4 in the first ratio but 1 in the second.

2. Scale the second ratio so that Q agrees: multiply Q : R = 1 : 2 throughout by 4, which gives Q : R = 4 : 8.

3. Merge the two into one chain: P : Q : R = 5 : 4 : 8.

4. Add the parts: 5 + 4 + 8 = 17 parts in all.

5. Find the value of one part: 884 ÷ 17 = 52 runs per part.

6. P holds 5 parts, so P scored 5 × 52 = 260 runs.

Cross-check by distributing all three shares:

The three shares add to 260 + 208 + 416 = 884, and 260 : 208 = 5 : 4 while 208 : 416 = 1 : 2, so both given ratios hold. P scored 260 runs.