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

Let's define:

Number of large shirts = $x$

Number of small shirts = $64 - x$ (since total shirts = 64)

Price of large shirt = $P$ (in INR)

Price of small shirt = $P - 50$ (since small shirt costs INR 50 less)

We're using the constraint that total shirts = 64, so if we have $x$ large shirts, we automatically have $(64-x)$ small shirts.

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From the problem:

Total cost of large shirts = INR 5000

Total cost of small shirts = INR 1800

This gives us two equations:

$P \times x = 5000$ ... (1)

$(P - 50) \times (64 - x) = 1800$ ... (2)

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From equation (1): $x = \frac{5000}{P}$

We can substitute this value of $x$ into equation (2) to get everything in terms of $P$.

$(P - 50) \times (64 - \frac{5000}{P}) = 1800$

$(P - 50) \times 64 - (P - 50) \times \frac{5000}{P} = 1800$

$64P - 3200 - \frac{5000(P - 50)}{P} = 1800$

$64P - 3200 - \frac{5000P - 250000}{P} = 1800$

$64P - 3200 - 5000 + \frac{250000}{P} = 1800$

$64P - 8200 + \frac{250000}{P} = 1800$

$64P + \frac{250000}{P} = 10000$

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$64P^2 + 250000 = 10000P$

$64P^2 - 10000P + 250000 = 0$

Dividing by 16: $4P^2 - 625P + 15625 = 0$

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We need to find two numbers that multiply to $(4 \times 15625 = 62500)$ and add to $-625$.

Those numbers are $-500$ and $-125$.

$4P^2 - 500P - 125P + 15625 = 0$

Factoring: $(4P - 125)(P - 125) = 0$

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From $(4P - 125)(P - 125) = 0$:

$P = \frac{125}{4} = 31.25$ or $P = 125$

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If $P = 31.25$:

Price of small shirt = $31.25 - 50 = -18.75$

This is impossible! Prices cannot be negative.

If $P = 125$:

Price of small shirt = $125 - 50 = 75$

This makes sense!

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Price of large shirt + Price of small shirt = $125 + 75 = 200$

Therefore, the price of a large shirt and a small shirt together is INR 200.