A man sells three motors for ₹ 5,400, ₹ 3,300 and ₹4,350 respectively. He makes 20% profit on the first and 10% profit on the second but on the whole he loses `9(3)/(8)%`.What did the third motor car cost him ?

Options

A. ₹6500
B. ₹6900
C. ₹5100
D. ₹7200

Explanation

To find the cost price of the third motor car, we can follow these steps:

### Step 1: Calculate the Selling Price (SP) of each motor
- SP of the first motor = ₹5,400
- SP of the second motor = ₹3,300
- SP of the third motor = ₹4,350

### Step 2: Calculate the Cost Price (CP) of the first motor
Given that the first motor is sold at a 20% profit:
- Let the CP of the first motor be X.
- Selling Price (SP) = CP + Profit
- 20% profit means SP = CP + 0.2 * CP = 1.2 * CP
- Therefore, 1.2 * X = 5,400
- To find X, we rearrange:
\[ X = \frac{5,400}{1.2} = 4,500 \]
So, the CP of the first motor is ₹4,500.

### Step 3: Calculate the Cost Price (CP) of the second motor
Given that the second motor is sold at a 10% profit:
- Let the CP of the second motor be Y.
- 10% profit means SP = CP + 0.1 * CP = 1.1 * CP
- Therefore, 1.1 * Y = 3,300
- To find Y, we rearrange:
\[ Y = \frac{3,300}{1.1} = 3,000 \]
So, the CP of the second motor is ₹3,000.

### Step 4: Calculate the total Selling Price (SP) of all motors
Total SP = SP of first motor + SP of second motor + SP of third motor
\[ \text{Total SP} = 5,400 + 3,300 + 4,350 = 13,050 \]

### Step 5: Calculate the overall loss percentage
The overall loss is given as \(9 \frac{3}{8}\%\), which can be converted to a fraction:
\[ 9 \frac{3}{8}\% = \frac{75}{8}\% = \frac{75}{800} \]
This means the loss is \(\frac{75}{800}\) of the total CP.

### Step 6: Set up the equation for total Cost Price (CP)
Let the total CP of all three motors be Z.
The loss can be expressed as:
\[ \text{Total SP} = \text{Total CP} - \text{Loss} \]
So, we can write:
\[ 13,050 = Z - \frac{75}{800}Z \]
This simplifies to:
\[ 13,050 = Z \left(1 - \frac{75}{800}\right) \]
Calculating \(1 - \frac{75}{800} = \frac{725}{800}\):
\[ 13,050 = Z \cdot \frac{725}{800} \]
To find Z, we rearrange:
\[ Z = 13,050 \cdot \frac{800}{725} \]
Calculating Z:
\[ Z = 13,050 \cdot \frac{800}{725} = 14,400 \]
So, the total CP of all three motors is ₹14,400.

### Step 7: Calculate the Cost Price (CP) of the third motor
Now, we know the CP of the first and second motors:
- CP of first motor = ₹4,500
- CP of second motor = ₹3,000

Let the CP of the third motor be R.
So, we have:
\[ 4,500 + 3,000 + R = 14,400 \]
This simplifies to:
\[ R = 14,400 - (4,500 + 3,000) = 14,400 - 7,500 = 6,900 \]

### Conclusion
The cost price of the third motor car is ₹6,900.

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