By selling 45 oranges for Rs. 40, a man loss `20%`. How many should he sell for Rs. 24 so as to gain `20%` in the transaction ?

Options

A. 16
B. 18
C. 20
D. 22

Answer

To solve the problem step by step, we need to follow these calculations:

### Step 1: Calculate the Cost Price (CP) of 45 oranges
Given that selling 45 oranges for Rs. 40 results in a loss of 20%, we can use the formula for loss:

\[ \text{Loss} = \text{Cost Price} - \text{Selling Price} \]

Since the loss is 20%, we can express this as:

\[ \text{Selling Price} = \text{Cost Price} \times \left(1 - \frac{\text{Loss \%}}{100}\right) \]

Substituting the values we have:

\[ 40 = \text{CP} \times \left(1 - \frac{20}{100}\right) \]

This simplifies to:

\[ 40 = \text{CP} \times 0.8 \]

Now, solving for CP:

\[ \text{CP} = \frac{40}{0.8} = 50 \]

### Step 2: Calculate the Cost Price (CP) of one orange
Now that we have the total cost price for 45 oranges, we can find the cost price of one orange:

\[ \text{CP of one orange} = \frac{\text{Total CP}}{\text{Number of oranges}} = \frac{50}{45} \approx 1.11 \]

### Step 3: Calculate the Selling Price (SP) for a gain of 20%
Next, we need to find out the selling price that would give a 20% gain. The formula for gain is:

\[ \text{Selling Price} = \text{Cost Price} \times \left(1 + \frac{\text{Gain \%}}{100}\right) \]

Using the cost price of one orange:

\[ \text{SP} = 1.11 \times \left(1 + \frac{20}{100}\right) = 1.11 \times 1.2 = 1.33 \]

### Step 4: Determine how many oranges should be sold for Rs. 24
Now, we need to find out how many oranges can be sold for Rs. 24 at the new selling price of Rs. 1.33 per orange:

\[ \text{Number of oranges} = \frac{\text{Total Selling Price}}{\text{SP per orange}} = \frac{24}{1.33} \approx 18 \]

Thus, the man should sell approximately **18 oranges** for Rs. 24 to gain 20%.

### Final Answer:
**18 oranges should be sold for Rs. 24 to gain 20%.**

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