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CAT 2019 Slot 2 — QA Question 2

Change in AverageEasy

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

Answer & solution

  • A

    4

  • B

    5

  • 4.5

  • D

    3.5

Solution

Easy

Fix the total from the overall average. To make the average of the 20 "small" integers as large as possible, make the other 10 integers as small as possible — and since they must exceed 55 and be integers, their smallest value is 66.

1

Total of all 30 integers. Average 55 over 3030 values.

Sum=30×5⇒ Sum=150\begin{aligned} &\text{Sum}=30\times 5\\ &\Rightarrow\ \text{Sum}=150 \end{aligned}
2

Make the other 10 integers minimal. Exactly 2020 integers do not exceed 55, so the remaining 1010 are greater than 55. The smallest integer greater than 55 is 66, so their minimum total is

Sum10≥10×6=60\begin{aligned} &\text{Sum}_{10}\ge 10\times 6=60 \end{aligned}
3

Maximise the 20 integers' total and average. Subtract the minimal total of step 2 from the grand total of step 1; the bigger leftover goes to the 2020 integers.

Sum20=150−60=90⇒ Average20=9020=4.5\begin{aligned} &\text{Sum}_{20}=150-60=90\\ &\Rightarrow\ \text{Average}_{20}=\frac{90}{20}=4.5 \end{aligned}
Highest possible average=4.5\text{Highest possible average}=4.5
CAT 2019 Slot 2 QA Q2: The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is — Solution | TheCATExam