Average and linear equations (sports scoring): Determine individual scores from linked conditions and the team average. Batsmen A, B, C, D, and E have an average score of 36 runs (total = 5 × 36 = 180). Given: D scored 5 more than E; E scored 8 fewer than A; B scored the total of D and E; and B + C = 107. Find the number of runs scored by E. Choose the correct value.

Given data

• Average of five batsmen = 36 ⇒ Total runs = 5 × 36 = 180.
• D = E + 5.
• E = A − 8 (i.e., A = E + 8).
• B = D + E.
• B + C = 107 ⇒ C = 107 − B.

Concept/Approach Form a single-variable equation in E by expressing A, B, C, D in terms of E and using the total 180.

Step-by-step calculation A = E + 8 D = E + 5 B = D + E = (E + 5) + E = 2E + 5 C = 107 − B = 107 − (2E + 5) = 102 − 2E Sum: A + B + C + D + E = (E + 8) + (2E + 5) + (102 − 2E) + (E + 5) + E = 3E + 120 Set equal to total: 3E + 120 = 180 ⇒ 3E = 60 ⇒ E = 20

Verification E = 20, D = 25, A = 28, B = 45, C = 62; total = 20 + 25 + 28 + 45 + 62 = 180 ✔

Common pitfalls Forgetting that “E scored 8 fewer than A” means A = E + 8 (not E = A + 8).