In a cricket match, five batsmen P, Q, R, S and T together score an average of 36 runs. Batsman S scores 5 runs more than T; T scores 8 runs fewer than P; Q scores as many runs as S and T together; and Q and R together score 107 runs. How many runs does T score?

Introduction / Context:
This is a multi-step word problem involving averages and several linear relationships among the runs scored by five batsmen. It tests logical translation of language into algebra and then systematic equation solving to determine the specific score of T, one of the batsmen.

Given Data / Assumptions:

• Batsmen: P, Q, R, S and T.
• The average score is 36 runs, so the total score is 5 × 36 = 180 runs.
• S scores 5 more than T, so S = T + 5.
• T scores 8 fewer than P, so T = P − 8, or P = T + 8.
• Q scores as many as S and T combined, so Q = S + T.
• Q and R together score 107 runs, so Q + R = 107.

Concept / Approach:
We express all players scores in terms of the single variable T and then use the total-sum condition to solve for T. This method eliminates the other variables one by one and ultimately leads to a simple linear equation in T, which is easy to solve.

Step-by-Step Solution:
From S = T + 5 and P = T + 8, the scores of P and S are expressed in terms of T.Q = S + T = (T + 5) + T = 2T + 5.From Q + R = 107, we get R = 107 − Q = 107 − (2T + 5) = 102 − 2T.Total runs: P + Q + R + S + T = 180. Substitute P, Q, R and S in terms of T: (T + 8) + (2T + 5) + (102 − 2T) + (T + 5) + T.Simplify: combine like terms to get 3T + 120 = 180, so 3T = 60 and T = 20.

Verification / Alternative check:
If T = 20, then S = 25, P = 28, Q = S + T = 45 and R = 107 − Q = 62.Check the total: P + Q + R + S + T = 28 + 45 + 62 + 25 + 20 = 180, which matches 5 × 36, so the data is consistent.

Why Other Options Are Wrong:
If T were 24, P would be 32, S 29, Q 53 and R 54, giving a total different from 180, so that option fails.Testing T = 28 or 29 similarly leads to totals that do not equal 180, so those values do not satisfy all the given conditions.

Common Pitfalls:
A common mistake is to misread "T scored 8 fewer than P" and write P = T − 8 instead of P = T + 8, which completely changes the equations.Another error is to forget that the average is over five batsmen, leading to a wrong total score and therefore wrong algebra.

Final Answer:
From the consistent system of equations, T must have scored 20 runs.