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CAT 2021 Slot 3 — QA Question 8

VariationEasy

One part of a hostel’s monthly expenses is fixed, and the other part is proportional to the number of its boarders. The hostel collects ₹ 1600 per month from each boarder. When the number of boarders is 50, the profit of the hostel is ₹ 200 per boarder, and when the number of boarders is 75, the profit of the hostel is ₹ 250 per boarder. When the number of boarders is 80, the total profit of the hostel, in INR, will be 

Answer & solution

  • A

    20000

  • B

    20200

  • C

    20800

  • 20500

Solution

Easy

Total cost == fixed cost FF plus a per-boarder cost kk times the number of boarders. Each boarder pays ₹1600\textsf{₹}1600, and profit per boarder is given for two boarder counts. Form two equations from total profit == revenue −- cost, solve for FF and kk, then compute profit at 8080 boarders.

1

Write the two profit equations. With bb boarders, revenue =1600b=1600b and cost =F+kb=F+kb. Total profit == (profit per boarder)×b\times b.

At b=50: 200(50)=1600(50)−(F+50k)...(1)At b=75: 250(75)=1600(75)−(F+75k)...(2)\begin{aligned} &\text{At }b=50:\ 200(50)=1600(50)-(F+50k) \quad\text{...(1)}\\ &\text{At }b=75:\ 250(75)=1600(75)-(F+75k) \quad\text{...(2)} \end{aligned}
2

Subtract to eliminate FF. Compute (2)−-(1).

18750−10000=1600(25)−25k[(2)-(1)]⇒ 8750=40000−25k⇒ 25k=31250 ⇒ k=1250\begin{aligned} &18750-10000=1600(25)-25k \quad\text{[(2)-(1)]}\\ &\Rightarrow\ 8750=40000-25k\\ &\Rightarrow\ 25k=31250\ \Rightarrow\ k=1250 \end{aligned}
3

Find the fixed cost. Substitute k=1250k=1250 into equation (1).

10000=80000−(F+50⋅1250)(from (1))⇒ 10000=80000−F−62500⇒ F=7500\begin{aligned} &10000=80000-(F+50\cdot 1250) \quad\text{(from (1))}\\ &\Rightarrow\ 10000=80000-F-62500\\ &\Rightarrow\ F=7500 \end{aligned}
4

Total profit at 8080 boarders. Profit == revenue −- cost with b=80b=80, F=7500F=7500, k=1250k=1250.

Profit=1600(80)−(7500+1250(80))⇒ =128000−(7500+100000)⇒ =128000−107500=20500\begin{aligned} &\text{Profit}=1600(80)-\big(7500+1250(80)\big)\\ &\Rightarrow\ =128000-(7500+100000)\\ &\Rightarrow\ =128000-107500=20500 \end{aligned}
Total profit=₹ 20500\text{Total profit}=\textsf{₹}\,20500
CAT 2021 Slot 3 QA Q8: One part of a hostel’s monthly expenses is fixed, and the other part is proportional to the number of it — Solution | TheCATExam