A person buys two watches for Rs. 1,000. He sells one at a loss of 5% and the other at 20% gain and on the whole he gains Rs. 50. Find the cost price of each watch.

Answer

To solve the problem step by step, let's denote the cost price of the first watch as \( x \) and the cost price of the second watch as \( y \).

### Step 1: Set up the equations
From the problem, we know that:
1. The total cost price of the two watches is Rs. 1000.
\[ x + y = 1000 \quad \text{(Equation 1)} \]

### Step 2: Calculate the selling prices
Next, we calculate the selling prices based on the given loss and gain percentages:
- The first watch is sold at a loss of 5%. Therefore, the selling price of the first watch is:
\[ \text{Selling Price of first watch} = x - 0.05x = 0.95x \]
- The second watch is sold at a gain of 20%. Therefore, the selling price of the second watch is:
\[ \text{Selling Price of second watch} = y + 0.20y = 1.20y \]

### Step 3: Set up the equation for total selling price
The total selling price of both watches can be expressed as:
\[ \text{Total Selling Price} = 0.95x + 1.20y \]
According to the problem, the overall gain is Rs. 50. Therefore, the total selling price is equal to the total cost price plus the gain:
\[ 0.95x + 1.20y = 1000 + 50 = 1050 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( x + y = 1000 \) (Equation 1)
2. \( 0.95x + 1.20y = 1050 \) (Equation 2)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 1000 - x \]

Now substitute \( y \) in Equation 2:
\[ 0.95x + 1.20(1000 - x) = 1050 \]
Expanding this gives:
\[ 0.95x + 1200 - 1.20x = 1050 \]
Combining like terms:
\[ -0.25x + 1200 = 1050 \]
Subtracting 1200 from both sides:
\[ -0.25x = 1050 - 1200 \]
\[ -0.25x = -150 \]
Dividing both sides by -0.25:
\[ x = \frac{-150}{-0.25} = 600 \]

### Step 5: Find the cost price of the second watch
Now substitute \( x \) back into Equation 1 to find \( y \):
\[ y = 1000 - 600 = 400 \]

### Conclusion
Thus, the cost price of the first watch is Rs. 600 and the cost price of the second watch is Rs. 400.

### Summary of Results
- Cost Price of First Watch = Rs. 600
- Cost Price of Second Watch = Rs. 400