CAT 2021

Slot 3

QA

Question & Solution

Let the number of large shirts be $l$ and the number of small shirts be $s$.

Let the price of a small shirt be $x$, and the price of a large shirt be $x + 50$.

We are given:

• $s + l = 64$
• $l(x + 50) = 5000$
• $sx = 1800$

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Add the two money equations:

$l(x + 50) + sx = 5000 + 1800 = 6800$

So: $$lx + sx + 50l = 6800$$

Using $s + l = 64$: $$64x + 50l = 6800$$

Now substitute: $$l = \frac{6800 - 64x}{50}$$

Plug this in the original equation: $$\frac{6800 - 64x}{50}(x + 50) = 5000$$

Multiply both sides by 50: $$(6800 - 64x)(x + 50) = 250000$$

Expand: $$6800x + 340000 - 64x^2 - 3200x = 250000$$

Rearrange: $$64x^2 - 3600x - 90000 = 0$$

Solve the quadratic: $$x = \frac{225 \pm 375}{8} = \frac{600}{8} \text{ or } -\frac{150}{8}$$

So the valid (positive) price is: $$x = 75$$

Thus:

• Small shirt price = $75$
• Large shirt price = $75 + 50 = 125$

Answer: $$75 + 125 = 200$$

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Alternate Approach (Using Options)

Each option represents:
Small + Large = Small + (Small + 50)

So,

• SMALL = (Option − 50)/2
• LARGE = SMALL + 50 = (Option + 50)/2

Options A and D give decimals for SMALL & LARGE, so check B and C.

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Option B:

• Large = $(150 + 50)/2 = 100$
• Small = $(150 - 50)/2 = 50$

Check total shirts: $$5000/100 + 1800/50 = 50 + 36 = 86 \quad \text{(Incorrect)}$$

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Option C:

• Large = $(200 + 50)/2 = 125$
• Small = $(200 - 50)/2 = 75$

Check: $$5000/125 + 1800/75 = 40 + 24 = 64 \quad \text{(Correct)}$$

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Final Answer: $\boxed{200}$