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CAT 2021 Slot 3 — QA Question 21

Solving Quadratic EquationsEasy

A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt. She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is

Answer & solution

  • A

    175

  • 200

  • C

    150

  • D

    225

Solution

Easy

Let the small shirt cost pp, so a large costs p+50p+50. The counts of each shirt are total-spend divided by unit price, and the two counts add to 6464. That gives one equation in pp, which is a quadratic.

1

Count equation. Large: 5000p+50\dfrac{5000}{p+50} shirts; small: 1800p\dfrac{1800}{p} shirts; together 6464.

5000p+50+1800p=64\begin{aligned} &\frac{5000}{p+50} + \frac{1800}{p} = 64 \end{aligned}
2

Clear denominators. Multiply through by p(p+50)p(p+50).

5000p+1800(p+50)=64 p(p+50)⇒ 6800p+90000=64p2+3200p⇒ 64p2−3600p−90000=0⇒ 4p2−225p−5625=0\begin{aligned} &5000p + 1800(p+50) = 64\,p(p+50)\\ &\Rightarrow\ 6800p + 90000 = 64p^2 + 3200p\\ &\Rightarrow\ 64p^2 - 3600p - 90000 = 0\\ &\Rightarrow\ 4p^2 - 225p - 5625 = 0 \end{aligned}
3

Solve and combine. Factor the quadratic.

(4p+75)(p−75)=0⇒ p=75(reject p=−754)⇒ small+large=75+(75+50)=200\begin{aligned} &(4p+75)(p-75)=0\\ &\Rightarrow\ p = 75 \quad\text{(reject }p=-\tfrac{75}{4})\\ &\Rightarrow\ \text{small}+\text{large} = 75 + (75+50) = 200 \end{aligned}
₹ 200₹\,200
CAT 2021 Slot 3 QA Q21: A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The — Solution | TheCATExam