CAT 2023 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like **Speed-time-distance**, **Mixture and Alligations**. This year, there was a surprise. The questions from **Geometry** were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 10 : 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

- 19
- 21
- 20
- 22

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Let the marks of each girl in the class be g and the marks of each boy be b.

Since, the average marks of 4 girls and 6 boys is 24…

4g + 6b = 24 × 10 = 240

A girl scores more than or equal to the score of a boy but never more than double the score.

Therefore, g = kb, where 1 ≤ k ≤ 2

4(kb) + 6b = 240

b(4k + 6) = 240

b(2k + 3) = 120

b = \( \frac { 120 } { 2 k + 3 } \)

Finally we need 2g + 6b = b(2k + 6) to be an integer.

\( 120 \times \left( \frac { 2 k + 6 } { 2 k + 3 } \right) \) needs to be an integer.

\( 120 \times \left( 1 + \frac { 3 } { 2 k + 3 } \right) \) needs to be an integer.

∴ \( 120 \times \left( \frac { 3 } { 2 k + 3 } \right) \) needs to be an integer.

\( \left( \frac { 360 } { 2 k + 3 } \right) \) needs to be an integer.

Let \( \frac { 360 } { 2 k + 3 } = n \), where n is an integer.

\( \frac { 360 } { n } - 3 = 2 k \)

Since, 1 ≤ k ≤ 2

2 ≤ 2k ≤ 4

2 ≤ \( \frac { 360 } { n } - 3 \) ≤ 4

5 ≤ \( \frac { 360 } { n } \) ≤ 7

\( \frac { 5 } { 360 } \) ≤ \( \frac { 1 } { n } \) ≤ \( \frac { 7 } { 360 } \)

\( \frac { 360 } { 7 } \) ≤ n ≤ \( \frac { 360 } { 5 } \)

51.42 ≤ n ≤ 72

52 ≤ n ≤ 72

Therefore, n can take 21 values, 52 to 72 both inclusive.

For all these values k takes a distinct value from 1 to 2 and 2g + 6b takes a distinct integral value.

Choice B is the correct answer.

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