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CAT 2020 Slot 1 — QA Question 9

Basics of AverageEasy

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

Answer & solution

  • A

    4

  • B

    5

  • 6

  • D

    7

Solution

Easy

Turn both word-statements into equations by clearing the fractions. Subtracting them removes cc and links AA and BB. The positive-integer constraint then pins the values down, but in fact A+BA+B is forced before you even need the exact split.

1

Form the two equations. "Sum of AA and the mean of B,CB,C is 5" and similarly for BB.

A+B+C2=5 ⇒ 2A+B+C=10(1)B+A+C2=7 ⇒ A+2B+C=14(2)\begin{aligned} &A+\frac{B+C}{2}=5\ \Rightarrow\ 2A+B+C=10 \quad\text{(1)}\\ &B+\frac{A+C}{2}=7\ \Rightarrow\ A+2B+C=14 \quad\text{(2)} \end{aligned}
2

Subtract to relate AA and BB. (2)−(1)(2)-(1) eliminates CC.

(A+2B+C)−(2A+B+C)=14−10⇒ B−A=4\begin{aligned} &(A+2B+C)-(2A+B+C)=14-10\\ &\Rightarrow\ B-A=4 \end{aligned}
3

Use positivity to fix the values. Put B=A+4B=A+4 into (1): 3A+C=63A+C=6 with A,C≥1A,C\ge 1. Only A=1A=1 works (then C=3C=3), giving B=5B=5.

3A+C=6, A≥1, C≥1 ⇒ A=1, C=3, B=5⇒ A+B=1+5=6\begin{aligned} &3A+C=6,\ A\ge1,\ C\ge1\ \Rightarrow\ A=1,\ C=3,\ B=5\\ &\Rightarrow\ A+B=1+5=6 \end{aligned}
A+B=6A+B=6
CAT 2020 Slot 1 QA Q9: Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, th — Solution | TheCATExam