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

Options

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

Answer

To solve the problem step-by-step, we will follow the information given in the question and use algebra to find the cost price of each clock.

### Step 1: Define Variables
Let the cost price of each clock be \( X \) rupees. Therefore, the total cost price for 90 clocks is:
\[ \text{Total Cost Price} = 90X \text{ rupees} \]

**Hint:** Start by defining a variable for the cost price to make calculations easier.

### Step 2: Calculate Selling Price for 40 Clocks at 10% Gain
The selling price for the 40 clocks sold at a gain of 10% is calculated as follows:
\[ \text{Selling Price of 40 Clocks} = 40 \times \left( X + \frac{10}{100}X \right) = 40 \times \left( \frac{110}{100}X \right) = 44X \text{ rupees} \]

**Hint:** Remember that selling price is calculated using the formula: Selling Price = Cost Price + Gain.

### Step 3: Calculate Selling Price for 50 Clocks at 20% Gain
The selling price for the 50 clocks sold at a gain of 20% is:
\[ \text{Selling Price of 50 Clocks} = 50 \times \left( X + \frac{20}{100}X \right) = 50 \times \left( \frac{120}{100}X \right) = 60X \text{ rupees} \]

**Hint:** Use the same formula for selling price for different percentages of gain.

### Step 4: Calculate Total Selling Price
Now, we can find the total selling price for all 90 clocks:
\[ \text{Total Selling Price} = \text{Selling Price of 40 Clocks} + \text{Selling Price of 50 Clocks} = 44X + 60X = 104X \text{ rupees} \]

**Hint:** Combine the selling prices to find the total selling price.

### Step 5: Calculate Selling Price if Sold at 15% Gain
If all clocks were sold at a uniform profit of 15%, the selling price would be:
\[ \text{Selling Price at 15% Gain} = 90 \times \left( X + \frac{15}{100}X \right) = 90 \times \left( \frac{115}{100}X \right) = 103.5X \text{ rupees} \]

**Hint:** Again, use the selling price formula for a uniform gain across all items.

### Step 6: Set Up the Equation
According to the problem, if sold at a uniform profit of 15%, he would have received 40 rupees less than what he actually received. Therefore, we can set up the equation:
\[ 104X = 103.5X + 40 \]

**Hint:** This step involves setting up an equation based on the information provided in the question.

### Step 7: Solve for X
Now, we will solve for \( X \):
\[ 104X - 103.5X = 40 \\ 0.5X = 40 \\ X = \frac{40}{0.5} = 80 \]

**Hint:** Isolate the variable on one side to find its value.

### Final Answer
The cost price of each clock is:
\[ \boxed{80} \text{ rupees} \]