By selling 45 lemons, a vendor loses a sum equal to the selling price of 3 lemons. Find his loss percent.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the cost price (CP) of one lemon be \( x \) and the selling price (SP) of one lemon be \( y \).

### Step 2: Calculate the Total Cost Price
The total cost price for 45 lemons is:
\[ \text{Total CP} = 45 \times x \]

### Step 3: Calculate the Total Selling Price
The total selling price for 45 lemons is:
\[ \text{Total SP} = 45 \times y \]

### Step 4: Determine the Loss
According to the problem, the vendor loses an amount equal to the selling price of 3 lemons. Therefore, the loss can be expressed as:
\[ \text{Loss} = \text{SP of 3 lemons} = 3y \]

### Step 5: Relate Loss to Cost Price and Selling Price
The loss can also be calculated as:
\[ \text{Loss} = \text{Total CP} - \text{Total SP} \]
Substituting the values we have:
\[ 3y = 45x - 45y \]

### Step 6: Simplify the Equation
Rearranging the equation gives:
\[ 3y + 45y = 45x \]
\[ 48y = 45x \]

### Step 7: Find the Selling Price in terms of Cost Price
From the equation \( 48y = 45x \), we can express \( y \) in terms of \( x \):
\[ y = \frac{45}{48}x = \frac{15}{16}x \]

### Step 8: Calculate the Loss
Now we can find the loss:
\[ \text{Loss} = \text{Total CP} - \text{Total SP} = 45x - 45y \]
Substituting \( y \):
\[ \text{Loss} = 45x - 45 \left(\frac{15}{16}x\right) \]
\[ = 45x - \frac{675}{16}x \]
To combine these, convert \( 45x \) into a fraction:
\[ = \frac{720}{16}x - \frac{675}{16}x = \frac{45}{16}x \]

### Step 9: Calculate the Loss Percentage
Loss percentage is calculated using the formula:
\[ \text{Loss Percentage} = \left(\frac{\text{Loss}}{\text{Total CP}}\right) \times 100 \]
Substituting the values:
\[ \text{Loss Percentage} = \left(\frac{\frac{45}{16}x}{45x}\right) \times 100 \]
\[ = \left(\frac{1}{16}\right) \times 100 = 6.25\% \]

### Final Answer
The loss percentage is **6.25%**.

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