TheCATExam
Sign in

CAT 2021 Slot 1 — QA Question 1

Weighted AverageEasy

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

Answer & solution

Answer: 35

Solution

Easy

Let the number admitted in A be the unknown. Express B's count and both averages in terms of it, then turn the "3 days more" condition into a single equation in one variable.

1

Set up the counts. Hospital A admitted 21 fewer patients than B, so if A admits xx patients, B admits x+21x+21.

A:x(let)⇒ B:x+21(21 more than A)\begin{aligned} &\text{A}: x \quad\text{(let)}\\ &\Rightarrow\ \text{B}: x+21 \quad\text{(21 more than A)} \end{aligned}
2

Write the two averages. Average recovery days =sum of recovery daysnumber of patients=\dfrac{\text{sum of recovery days}}{\text{number of patients}}, with totals 200200 for A and 152152 for B.

avgA=200xavgB=152x+21\begin{aligned} &\text{avg}_A=\frac{200}{x}\\ &\text{avg}_B=\frac{152}{x+21} \end{aligned}
3

Apply the "3 more" condition. A's average exceeds B's by 33 days.

200x=152x+21+3(from step 2)⇒ 200(x+21)=152x+3x(x+21)(multiply by x(x+21))⇒ 200x+4200=152x+3x2+63x⇒ 3x2+15x−4200=0⇒ x2+5x−1400=0(divide by 3)\begin{aligned} &\frac{200}{x}=\frac{152}{x+21}+3 \quad\text{(from step 2)}\\ &\Rightarrow\ 200(x+21)=152x+3x(x+21) \quad\text{(multiply by }x(x+21)\text{)}\\ &\Rightarrow\ 200x+4200=152x+3x^{2}+63x\\ &\Rightarrow\ 3x^{2}+15x-4200=0\\ &\Rightarrow\ x^{2}+5x-1400=0 \quad\text{(divide by 3)} \end{aligned}
4

Solve and discard the impossible root. Factorising x2+5x−1400=(x−35)(x+40)x^{2}+5x-1400=(x-35)(x+40).

(x−35)(x+40)=0⇒ x=35  or  x=−40(count cannot be negative)⇒ x=35\begin{aligned} &(x-35)(x+40)=0\\ &\Rightarrow\ x=35 \ \text{ or } \ x=-40 \quad\text{(count cannot be negative)}\\ &\Rightarrow\ x=35 \end{aligned}
Number admitted in hospital A=35\text{Number admitted in hospital A}=35
CAT 2021 Slot 1 QA Q1: Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The — Solution | TheCATExam