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 in INR, is

Let the price of a large shirt be \( x \) INR and the price of a small shirt be \( y \) INR. From the problem, we know that:

1. The price of a small shirt is INR 50 less than that of a large shirt, so:
   \[
   y = x - 50
   \]

2. The total amount paid for the large shirts is INR 5000, and for the small shirts is INR 1800. If \( L \) is the number of large shirts and \( S \) is the number of small shirts, we have the following equations:
   \[
   L \cdot x = 5000 \quad \text{(for large shirts)}
   \]
   \[
   S \cdot y = 1800 \quad \text{(for small shirts)}
   \]

3. The total number of shirts is 64, so:
   \[
   L + S = 64
   \]

Now, we can solve the system of equations:

### Step 1: Express \( L \) and \( S \) in terms of \( x \) and \( y \)
From the first two equations:
   \[
   L = \frac{5000}{x} \quad \text{and} \quad S = \frac{1800}{y}
   \]

Substitute \( y = x - 50 \) into the second equation for \( S \):
   \[
   S = \frac{1800}{x - 50}
   \]

### Step 2: Use the total number of shirts equation
Now substitute \( L \) and \( S \) into \( L + S = 64 \):
   \[
   \frac{5000}{x} + \frac{1800}{x - 50} = 64
   \]

### Step 3: Solve the equation
To solve this, we can find a common denominator and solve for \( x \):
   \[
   \frac{5000(x - 50) + 1800x}{x(x - 50)} = 64
   \]

Simplifying the numerator:
   \[
   5000x - 250000 + 1800x = 6800x - 250000
   \]

Thus, the equation becomes:
   \[
   \frac{6800x - 250000}{x(x - 50)} = 64
   \]

Multiply both sides by \( x(x - 50) \) to eliminate the denominator:
   \[
   6800x - 250000 = 64x(x - 50)
   \]

Expand the right-hand side:
   \[
   6800x - 250000 = 64x^2 - 3200x
   \]

Rearrange the equation to form a quadratic equation:
   \[
   64x^2 - 10000x + 250000 = 0
   \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula:
\[
x = \frac{-(-10000) \pm \sqrt{(-10000)^2 - 4(64)(250000)}}{2(64)}
\]

Simplifying the discriminant:
\[
x = \frac{10000 \pm \sqrt{100000000 - 64000000}}{128}
\]
\[
x = \frac{10000 \pm \sqrt{36000000}}{128}
\]
\[
x = \frac{10000 \pm 6000}{128}
\]

This gives two solutions:
\[
x = \frac{10000 + 6000}{128} = \frac{16000}{128} = 125 \quad \text{or} \quad x = \frac{10000 - 6000}{128} = \frac{4000}{128} = 31.25
\]

Since \( x \) represents the price of a large shirt, it must be a reasonable whole number. So, \( x = 125 \).

### Step 5: Find the price of a small shirt
Since \( y = x - 50 \), we have:
\[
y = 125 - 50 = 75
\]

### Final Answer:
The price of a large shirt and a small shirt together is:
\[
x + y = 125 + 75 = 200 \text{ INR}
\]

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

---

Would you like more details or have any questions? Here are 5 related questions to explore:

1. How would the solution change if the price difference between shirts was INR 100?
2. What if the total number of shirts bought was 80 instead of 64?
3. How can you verify that the total paid for the shirts is correct given the prices?
4. What happens if the total cost for the large shirts increases by 10%?
5. Can this problem be solved using a graphical method instead of algebraic?

**Tip:** Always cross-check your equations when dealing with systems to ensure no steps are missed!

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

Learn how to solve a system of equations to determine the price of large and small shirts bought by a shop owner. The problem involves using algebra, quadratic equations, and systems of equations to find that the total price of both shirts is INR 200. Suitable for students in Grades 9-12.

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