Average, Mixture and Alligation Question 4 - CAT 2022

Step-by-Step Explanation

Let the number of students in section B be x. The number of students in section A is given as x − 10.

• Total marks in section A = 32(x − 10)
• Total marks in section B = 60x

Set up the equation for the combined average marks across both sections:

Combined Average = [32(x − 10) + 60x] / [(x − 10) + x] = (92x − 320) / (2x − 10)

Simplify the fraction by dividing the numerator and denominator by 2:

Combined Average = (46x − 160) / (x − 5)

Perform algebraic polynomial division to separate the variable from the integer baseline:

(46(x − 5)+70) / (x − 5) = 46 + 70 / (x − 5)

For the combined average to be a clean integer, the term x − 5 must be a perfectly dividing factor of 70.

Additionally, the count of students in section A (x − 10) must be greater than zero, meaning x > 10, which implies our divisor x − 5 > 5.

The factors of 70 that are strictly greater than 5 are: 7, 10, 14, 35, and 70.

• To minimize students in A: Choose the smallest valid factor, x − 5 = 7 ⟹ x = 12.

Minimum students in A = 12 − 10 = 2

• To maximize students in A: Choose the largest valid factor, x − 5 = 70 ⟹ x = 75.

Maximum students in A = 75 − 10 = 65

The difference between the maximum and minimum number of students in section A is:

Difference = 65 − 2 = 63

Correct Answer: 63

💡 FundaMakers Tip:

Whenever you face an algebraic fraction required to be an integer, always rewrite the numerator in terms of the denominator (e.g., transforming 46x − 160 into 46x − 5 + 70). This isolate-and-conquer strategy turns a guessing game into a simple task of picking standard factors of a constant number, instantly establishing clear boundaries.