A person purchases 90 clocks and sells 40 clocks at a gain of 10% and 50 clocks at gain of 20%. If he sold all of them at a uniform profit of 15%, then he would have got Rs. 40 less. The cost price of each clock is:

Options

A. Rs. 50
B. Rs. 60
C. Rs, 80
D. Rs. 90

Answer

To solve the problem step by step, we will first define the variables and then calculate the cost price of each clock based on the given information.

### Step 1: Define the Variables
Let the cost price of each clock be \( x \).

### Step 2: Calculate the Total Cost Price
The total cost price for 90 clocks will be:
\[ \text{Total Cost Price} = 90 \times x = 90x \]

### Step 3: Calculate Selling Price for 40 Clocks at 10% Gain
For the 40 clocks sold at a gain of 10%, the selling price can be calculated as:
\[ \text{Selling Price of 40 clocks} = \text{Cost Price} + \text{Gain} = 40x + 0.10 \times 40x = 40x + 4x = 44x \]

### Step 4: Calculate Selling Price for 50 Clocks at 20% Gain
For the 50 clocks sold at a gain of 20%, the selling price is:
\[ \text{Selling Price of 50 clocks} = \text{Cost Price} + \text{Gain} = 50x + 0.20 \times 50x = 50x + 10x = 60x \]

### Step 5: Calculate Total Selling Price
Now, we can calculate the total selling price for all 90 clocks:
\[ \text{Total Selling Price} = \text{Selling Price of 40 clocks} + \text{Selling Price of 50 clocks} = 44x + 60x = 104x \]

### Step 6: Calculate Selling Price if Sold at Uniform Profit of 15%
If all clocks were sold at a uniform profit of 15%, the selling price would be:
\[ \text{Selling Price at 15% profit} = \text{Total Cost Price} + \text{Gain} = 90x + 0.15 \times 90x = 90x + 13.5x = 103.5x \]

### Step 7: Set Up the Equation
According to the problem, if he sold all clocks at a uniform profit of 15%, he would have received Rs. 40 less than the actual selling price. Therefore, we can set up the equation:
\[ 104x = 103.5x + 40 \]

### Step 8: Solve for \( x \)
Now, we can solve for \( x \):
\[ 104x - 103.5x = 40 \]
\[ 0.5x = 40 \]
\[ x = \frac{40}{0.5} = 80 \]

### Conclusion
The cost price of each clock is:
\[ \boxed{80} \]