In a Mathematics examination the number scored by 5 candidates are 5 successive odd integers. If their total marks are 185, the highest score is

Concept & Strategy For an odd number of consecutive integers (even or odd), the average of the set is exactly equal to the middle integer. $$ \text{Middle Term} = \frac{\text{Sum}}{\text{Number of Terms}} $$ Step-by-Step Solution * **Given:** 5 candidates scored successive odd integers. Total sum = 185. * **Calculation:** Let the 5 consecutive odd integers be $(x - 4), (x - 2), x, (x + 2), (x + 4)$. * Sum these terms: $(x - 4) + (x - 2) + x + (x + 2) + (x + 4) = 185$ * Notice how the constants cancel out perfectly: $5x = 185$ * Solve for $x$ (the middle score): $x = 185 / 5 = 37$ * The terms are $33, 35, 37, 39, 41$. The question asks for the highest score, which is $(x + 4)$. $\text{Highest Score} = 37 + 4 = 41$ Exam Strategy & Shortcut **Average equals Middle Term:** Immediately divide the sum by the number of terms to find the middle number. $185 / 5 = 37$. Because there are 5 terms, there are two terms above the middle. The consecutive odd numbers above 37 are 39 and 41. Thus, 41 is the highest. This can be done entirely mentally. Common Pitfall Setting the integers as $x, x+1, x+2...$ instead of $x, x+2, x+4...$. Successive odd (or even) integers are separated by 2, not 1. Final Answer **Therefore, the correct answer is 41.**