TheCATExam
Sign in

CAT 2023 Slot 2 — QA Question 16

DiscountEasy

Jayant bought a certain number of white shirts at the rate of Rs 1000 per piece and a certain number of blue shirts at the rate of Rs 1125 per piece. For each shirt, he then set a fixed market price which was 25% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10% and made a total profit of Rs 51000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is

Answer & solution

Answer: 407

Solution

Easy

Mark-up then discount act on the average cost, so the overall multiplier on total cost is 1.25×0.9=1.1251.25\times0.9=1.125 — a flat 12.5%12.5\% profit. That turns the profit equation into one linear relation between the white count ww and blue count bb. To maximise w+bw+b, load up on the cheaper-coefficient shirt and use the fewest of the other (but at least one of each).

1

Selling price as a multiple of cost. Marked price =1.25×=1.25\times average cost, sold at 10%10\% off:

Multiplier=1.25×0.9=1.125Profit rate=1.125−1=0.125\begin{aligned} &\text{Multiplier}=1.25\times0.9=1.125\\ &\text{Profit rate}=1.125-1=0.125 \end{aligned}
2

Profit equation on total cost 1000w+1125b1000w+1125b:

0.125 (1000w+1125b)=510001000w+1125b=408000\begin{aligned} &0.125\,(1000w+1125b)=51000\\ &1000w+1125b=408000 \end{aligned}
3

Simplify (divide by 2525, then by 55):

40w+45b=163208w+9b=3264\begin{aligned} &40w+45b=16320\\ &8w+9b=3264 \end{aligned}
4

Maximise w+bw+b. Keep bb (bigger coefficient) as small as possible while staying a positive integer with ww also a positive integer. From 8w=3264−9b8w=3264-9b, we need 3264−9b3264-9b divisible by 88. Since 32643264 is divisible by 88, we need 9b9b — hence bb — divisible by 88. The smallest such b≥1b\ge 1 is b=8b=8:

b=8⇒ 8w=3264−72=3192⇒ w=399w+b=399+8=407\begin{aligned} &b=8\Rightarrow\ 8w=3264-72=3192\Rightarrow\ w=399\\ &w+b=399+8=407 \end{aligned}

The maximum possible total number of shirts is 407\mathbf{407}.

Why b=1b=1 fails: 8w+9=3264⇒8w=32558w+9=3264\Rightarrow 8w=3255, not a multiple of 88. Stepping bb up, the first value making 8w8w an integer (and w>0w>0) is b=8b=8, which also maximises w+bw+b.

CAT 2023 Slot 2 QA Q16: Jayant bought a certain number of white shirts at the rate of Rs 1000 per piece and a certain number of blue s — Solution | TheCATExam