By selling 32 oranges for Rs 30, a man loses 25%. How many oranges should be sold for Rs 24, so as to gain 20% in the transaction?

Options

A. 20
B. 22
C. 16
D. 19

Answer

To solve the problem step by step, we will first find the cost price (CP) of the oranges and then determine how many oranges need to be sold for Rs 24 to achieve a 20% gain.

### Step 1: Find the Cost Price (CP) of 32 Oranges
Given that 32 oranges are sold for Rs 30 with a loss of 25%, we can use the formula for selling price (SP) when there is a loss:

\[ SP = (1 - \text{Loss Percentage}) \times CP \]

Here, the loss percentage is 25%, so:

\[ SP = (1 - 0.25) \times CP = 0.75 \times CP \]

We know that the selling price (SP) is Rs 30:

\[ 0.75 \times CP = 30 \]

Now, we can solve for CP:

\[ CP = \frac{30}{0.75} = 40 \text{ Rs} \]

### Step 2: Determine the Cost Price of One Orange
Since the cost price of 32 oranges is Rs 40, the cost price of one orange is:

\[ \text{Cost Price of One Orange} = \frac{40}{32} = 1.25 \text{ Rs} \]

### Step 3: Calculate the Cost Price (CP) for Selling at Rs 24 with a 20% Gain
Now, we need to find out how many oranges should be sold for Rs 24 to gain 20%. The formula for selling price (SP) when there is a gain is:

\[ SP = (1 + \text{Gain Percentage}) \times CP \]

Here, the gain percentage is 20%, so:

\[ SP = (1 + 0.20) \times CP = 1.20 \times CP \]

We know that the selling price (SP) is Rs 24:

\[ 1.20 \times CP = 24 \]

Now, we can solve for CP:

\[ CP = \frac{24}{1.20} = 20 \text{ Rs} \]

### Step 4: Find the Number of Oranges for Rs 20 Cost Price
Now we know that the cost price of 32 oranges is Rs 40. We need to find how many oranges correspond to a cost price of Rs 20.

Using the ratio:

\[ \text{If } 32 \text{ oranges cost } 40 \text{ Rs, then } x \text{ oranges cost } 20 \text{ Rs} \]

We can set up the proportion:

\[ \frac{32}{40} = \frac{x}{20} \]

Cross-multiplying gives:

\[ 32 \times 20 = 40 \times x \]

\[ 640 = 40x \]

Now, solving for x:

\[ x = \frac{640}{40} = 16 \]

### Conclusion
The number of oranges that should be sold for Rs 24 to gain a 20% profit is **16 oranges**.

---