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CAT 2023 Slot 1 — QA Question 17

Basics of AverageEasy

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

Answer & solution

  • A

    20

  • B

    19

  • 21

  • D

    22

Solution

Easy

Translate the average into one linear equation in the per-girl mark gg and per-boy mark bb, then turn the two inequalities ("a girl scores between one and two boys") into a range. The quantity asked for, 2g+6b2g+6b, conveniently rewrites in terms of bb alone — count the integers in its range.

1

Turn the average into an equation. With 44 girls and 66 boys averaging 2424:

4g+6b10=24 ⇒ 4g+6b=240 ⇒ 2g+3b=120(1)\frac{4g+6b}{10}=24\ \Rightarrow\ 4g+6b=240\ \Rightarrow\ 2g+3b=120 \quad(1)
2

Encode the constraint b≤g≤2bb\le g\le 2b in the same form as (1)(1). Double it, then add 3b3b:

b≤g≤2b ⇒ 2b≤2g≤4b⇒ 5b≤2g+3b⏟=120≤7b(2)\begin{aligned} &b\le g\le 2b\ \Rightarrow\ 2b\le 2g\le 4b\\ &\Rightarrow\ 5b\le \underbrace{2g+3b}_{=120}\le 7b \quad(2) \end{aligned}
3

Bound bb. From (2)(2) with 2g+3b=1202g+3b=120:

5b≤120 and 120≤7b ⇒ 1207≤b≤24 ⇒ 1717≤b≤245b\le 120\ \text{and}\ 120\le 7b\ \Rightarrow\ \frac{120}{7}\le b\le 24\ \Rightarrow\ 17\tfrac17\le b\le 24
4

Rewrite the target. The total for 22 girls and 66 boys is

2g+6b=(2g+3b)+3b=120+3b2g+6b=(2g+3b)+3b=120+3b

so its range follows from the bound on bb:

120+3⋅1207≤120+3b≤120+3⋅24 ⇒ 171.43≤2g+6b≤192120+3\cdot\tfrac{120}{7}\le 120+3b\le 120+3\cdot24\ \Rightarrow\ 171.43\le 2g+6b\le 192
5

Count the integers in [171.43, 192][171.43,\,192] — these are 172,173,…,192172,173,\dots,192:

192−172+1=21192-172+1=21

There are 21\mathbf{21} distinct possible integer totals.

bb and gg themselves need not be integers — only the total 2g+6b=120+3b2g+6b=120+3b must be a whole number, and it sweeps continuously across (171.43, 192](171.43,\,192], so every integer in that window is achievable.

CAT 2023 Slot 1 QA Q17: In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while e — Solution | TheCATExam