In an engineering college the average salary of all engineering graduates from Mechanical trade is ₹ 2.45 lacs per annum and that of the engineering graduates from Electronics trade is ₹ 3.56 lacs per annum. The average salary of all Mechanical and Electronics graduates is ₹ 3.12 lacs per annum. Find the least number of Electronics graduates passing out from this institute.

### Concept & Logic This problem uses the **Rule of Alligation** to find the ratio of graduates in the two trades. The crucial secondary concept is understanding that the number of graduates must be a whole integer. Finding the "least number" requires simplifying the resulting ratio to its lowest coprime terms. ### Step-by-Step Solution * **Given Data:** Mechanical Average = $2.45$ lacs Electronics Average = $3.56$ lacs Combined Average = $3.12$ lacs * **Applying Alligation:** Mech ($2.45$) ------------- Elec ($3.56$) \ / Mean ($3.12$) / \ $(3.56 - 3.12)$ ----------- $(3.12 - 2.45)$ Ratio of Mech : Elec = $0.44 : 0.67$ Multiply both sides by 100 to remove decimals: Ratio of Mech : Elec = $44 : 67$ * **Deducing the Least Number:** The ratio of Mechanical to Electronics graduates is $44 : 67$. This means for every $44$ Mechanical graduates, there are $67$ Electronics graduates. To find the *least* possible actual number of graduates, we check if the ratio $44/67$ can be simplified further. The number $67$ is a prime number, meaning $44$ and $67$ are coprime (they share no common factors other than $1$). Therefore, the ratio cannot be reduced to smaller whole numbers. The smallest integer values that satisfy this ratio are $44$ Mechanical and $67$ Electronics graduates. ### Exam Strategy & Shortcut Perform the cross-subtraction directly mentally: $356 - 312 = 44$ (Mech parts) $312 - 245 = 67$ (Elec parts) Ratio is $44 : 67$. Recognizing immediately that $67$ is prime confirms that the ratio is fully simplified, locking in $67$ as the absolute minimum number of Electronics students. ### Common Pitfall The primary trap here is option (d) "Cannot be determined". Because the exact total number of students isn't provided, students quickly assume there is missing information. The key lies in the word "least". While the *actual* number could be 134 ($67 \times 2$) or 201 ($67 \times 3$), the *least possible* number is the prime ratio base itself. ### Final Answer **Therefore, the correct answer is 67.**