CAT 2023 Slot 1 QA Q10 Ratio, Mixtures & Averages

Question 10 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

A
19
B
21
C
20
D
22

Answer: (B) 21 Ratio, Mixtures & Averages MCQ

Solution

Let each girl score \(g\), each boy \(b\): \(4g + 6b = 240\), so \(g = 60 - 1.5b\).
\(g \ge b\) gives \(b \le 24\); \(g \le 2b\) gives \(60 \le 3.5b\), i.e. \(b \ge \frac{120}{7} \approx 17.14\).
Required total \(2g + 6b = 120 - 3b + 6b = 120 + 3b\), with \(3b \in [51.43, 72]\).
Integer totals: \(3b = 52, 53, \dots, 72\), i.e. 21 values.