A car agency has 108 cars. He sold some cars at 9% profit and rest at 36% profit. Thus he gains 17% on the sale of all his cars. The no. of cars sold at 36% profit is :

Options

A. 25
B. 32
C. 35
D. 75

Answer

To solve the problem, we will use the concept of alligation to find the number of cars sold at 36% profit.

Let:
- \( x \) = number of cars sold at 9% profit
- \( y \) = number of cars sold at 36% profit

From the problem, we know:
1. The total number of cars is 108:
\[ x + y = 108 \]

2. The overall profit is 17%. We can express the profit in terms of the profits from the two groups of cars:
- Profit from cars sold at 9% = \( 0.09x \)
- Profit from cars sold at 36% = \( 0.36y \)
- Total profit from all cars = \( 0.17 \times 108 \)

Therefore, we can set up the equation:
\[ 0.09x + 0.36y = 0.17 \times 108 \]

Now, let's calculate \( 0.17 \times 108 \):
\[ 0.17 \times 108 = 18.36 \]

So, we have the equation:
\[ 0.09x + 0.36y = 18.36 \]

Now we have a system of equations:
1. \( x + y = 108 \)
2. \( 0.09x + 0.36y = 18.36 \)

### Step 1: Solve for \( x \) in terms of \( y \)
From the first equation, we can express \( x \) as:
\[ x = 108 - y \]

### Step 2: Substitute \( x \) in the second equation
Substituting \( x \) in the second equation gives:
\[ 0.09(108 - y) + 0.36y = 18.36 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 0.09 \times 108 - 0.09y + 0.36y = 18.36 \]
Calculating \( 0.09 \times 108 \):
\[ 0.09 \times 108 = 9.72 \]
So, we have:
\[ 9.72 - 0.09y + 0.36y = 18.36 \]
Combining like terms:
\[ 9.72 + 0.27y = 18.36 \]

### Step 4: Isolate \( y \)
Subtract \( 9.72 \) from both sides:
\[ 0.27y = 18.36 - 9.72 \]
Calculating \( 18.36 - 9.72 \):
\[ 0.27y = 8.64 \]

### Step 5: Solve for \( y \)
Dividing both sides by \( 0.27 \):
\[ y = \frac{8.64}{0.27} = 32 \]

### Step 6: Find \( x \)
Now, substitute \( y \) back into the equation for \( x \):
\[ x = 108 - 32 = 76 \]

### Conclusion
The number of cars sold at 36% profit is \( y = 32 \).