Average score of a class of 60 students, in an exam, was 43. Average score of the students who had passed is 52 and the average score of students who had failed is 16. How many failed the exam?

### Concept & Strategy This problem is best solved using the **Rule of Alligation**. Alligation helps us quickly find the ratio in which two distinct groups (passed and failed students) are mixed to form a combined group (the whole class) with a known overall average. $$ \frac{\text{Quantity of Group 1}}{\text{Quantity of Group 2}} = \frac{\text{Average of Group 2} - \text{Overall Average}}{\text{Overall Average} - \text{Average of Group 1}} $$ ### Step-by-Step Solution * **Given:** * Total students = $60$ * Overall average score = $43$ * Average score of passed students = $52$ * Average score of failed students = $16$ * **Calculation / Deduction:** 1. Set up the alligation cross: Passed Average ($52$) -------------- Failed Average ($16$) \ / \ / Overall ($43$) / \ / \ ($43 - 16$) ------------------------ ($52 - 43$) $= 27$ $= 9$ 2. Find the ratio of Passed to Failed students: Ratio = $27 : 9 = 3 : 1$ 3. Calculate the number of failed students: The total number of parts is $3 + 1 = 4$ parts. $4$ parts correspond to the total class size of $60$ students. $1$ part = $\frac{60}{4} = 15$ students. Since the ratio of failed students is $1$ part, the number of failed students is $15$. ### Exam Strategy & Shortcut Alligation is the ultimate shortcut for "mixture" average problems. If you used the standard algebraic equation $52x + 16(60-x) = 43 \times 60$, you would waste precious seconds multiplying and shifting large numbers. The alligation cross gives you the $3:1$ ratio almost instantly, turning a complex multiplication problem into a simple division problem ($60 \div 4$). ### Common Pitfall A frequent mistake is matching the final ratio to the wrong group. In the alligation cross, the difference calculated on the left side corresponds to the quantity of the group on the right side. Always remember to cross the values properly so you don't accidentally conclude that $45$ students failed. ### Final Answer Therefore, the correct answer is **15**.