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 step by step, we need to use the concept of alligation. Here’s how we can break it down:

### Step 1: Identify the given information
- Total number of cars = 108
- Profit on some cars = 9%
- Profit on the rest of the cars = 36%
- Overall profit = 17%

### Step 2: Set up the alligation
We can set up the alligation to find the ratio of the number of cars sold at different profit percentages. The formula for alligation is:

\[ \text{Profit of higher group} - \text{Overall profit} : \text{Overall profit} - \text{Profit of lower group} \]

Substituting the values we have:
- Profit of higher group (36%) = 36%
- Overall profit = 17%
- Profit of lower group (9%) = 9%

Calculating the differences:
- \( 36\% - 17\% = 19\% \)
- \( 17\% - 9\% = 8\% \)

So, the ratio of cars sold at 36% profit to those sold at 9% profit is:
\[ 19 : 8 \]

### Step 3: Let the number of cars sold at 9% profit be \( 19x \) and at 36% profit be \( 8x \)
From the ratio, we can express the number of cars sold at each profit level:
- Cars sold at 9% profit = \( 19x \)
- Cars sold at 36% profit = \( 8x \)

### Step 4: Set up the equation based on total cars
The total number of cars sold is equal to the total cars available:
\[ 19x + 8x = 108 \]
\[ 27x = 108 \]

### Step 5: Solve for \( x \)
Dividing both sides by 27:
\[ x = \frac{108}{27} = 4 \]

### Step 6: Calculate the number of cars sold at 36% profit
Now, we can find the number of cars sold at 36% profit:
\[ \text{Cars sold at 36% profit} = 8x = 8 \times 4 = 32 \]

### Final Answer
The number of cars sold at 36% profit is **32**.

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