The average score of a batsman in twenty matches is 15 and in twenty-five other matches is 24. Find the average score in all the forty-five matches.

Correct — B. This is a weighted average: add the runs, add the matches, then divide once. Averaging the two averages is not the same sum.

Runs in the first twenty = 20 × 15 = 300
Runs in the next twenty-five = 25 × 24 = 600
Total runs = 300 + 600 = 900

Total matches = 20 + 25 = 45
Average = 900 ÷ 45 = 20 → option (b).

Why the others are wrong

• (a)
  21 — 21 needs a different split of matches. It would take the higher-scoring set to be twice the size of the lower — 20 matches at 15 and 40 at 24 average 21 — but here the two sets are 20 and 25.
• (c)
  19.5 — The unweighted mean. (15 + 24) ÷ 2 = 19.5 averages the two averages and ignores that the 24 was scored across twenty-five matches while the 15 covers only twenty.
• (d)
  22 — More runs than he scored. An average of 22 over 45 matches means 990 runs; the innings here total 300 + 600 = 900, so 20 is the ceiling as well as the answer.

A weighted average is total quantity ÷ total count. The plain mean of two averages is only correct when the two groups are the same size, and here they are not — twenty against twenty-five.

The unequal weights pull the result: the answer must fall between 15 and 24, and closer to 24 because the larger group scored 24. The midpoint 19.5 is below 20, and the true value sits just above it.

Alligation gives the same number faster. The distances 20 − 15 = 5 and 24 − 20 = 4 are in the ratio 5 : 4, the reverse of the match counts 20 : 25 = 4 : 5.

Key facts

• A weighted average is the total divided by the total count, not the mean of the component averages.
• 20 matches at 15 give 300 runs and 25 matches at 24 give 600 runs, making 900 runs in 45 matches.
• The unweighted mean of 15 and 24 is 19.5, and it appears among the printed options.
• A weighted average always lies between the smallest and largest group averages, so it had to fall between 15 and 24.

Common traps

• Averaging the two averages, which gives 19.5 instead of 20
• Dividing by 2 rather than by the 45 matches
• Reading twenty-five other matches as a total of twenty-five instead of twenty-five more

No directly related past PYQ was found.