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CAT 2017 Slot 1 — QA Question 14

AlligationEasy

A class consists of 20 boys and 30 girls. In the mid-semester examination, the average score of the girls was 5 higher than that of the boys. In the final exam, however, the average score of the girls dropped by 3 while the average score of the entire class increased by 2. The increase in the average score of the boys is:

Answer & solution

Correct answer: 9.5

  • 9.5

  • B

    10

  • C

    4.5

  • D

    6

Solution

Easy

Let the boys' mid-sem average be bb. Write the class average as a weighted mean of 2020 boys and 3030 girls, apply the stated final-exam changes, and solve for the new boys' average.

1

Mid-semester class average. Girls average b+5b+5; weight by counts 2020 and 3030:

Cˉ=20b+30(b+5)50=50b+15050=b+3\begin{aligned} &\bar{C}=\frac{20b+30(b+5)}{50}=\frac{50b+150}{50}=b+3 \end{aligned}
2

Apply the final-exam changes. Girls drop by 33; class average rises by 22. Let new boys' average be xx.

new girls=(b+5)−3=b+2new class=(b+3)+2=b+5(from step 1)\begin{aligned} &\text{new girls}=(b+5)-3=b+2\\ &\text{new class}=(b+3)+2=b+5 \quad\text{(from step 1)} \end{aligned}
3

Solve for the new boys' average. Weighted mean of the final scores equals b+5b+5:

20x+30(b+2)50=b+5⇒ 20x+30b+60=50b+250⇒ 20x=20b+190⇒ x=b+9.5\begin{aligned} &\frac{20x+30(b+2)}{50}=b+5\\ &\Rightarrow\ 20x+30b+60=50b+250\\ &\Rightarrow\ 20x=20b+190\\ &\Rightarrow\ x=b+9.5 \end{aligned}
4

Read off the increase. The boys' average rose from bb to b+9.5b+9.5.

x−b=9.5\begin{aligned} &x-b=9.5 \end{aligned}
Increase=9.5\text{Increase}=9.5

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CAT 2017 Slot 1 QA Q14: A class consists of 20 boys and 30 girls. In the mid-semester examination, the average score of the girls was — Solution | TheCATExam