I loss 9 per cent by selling pencils at the rate of 15 a rupee. How many for a rupee must I sell them to gain 5 per cent?

Options

A. 10
B. 13
C. 15
D. 18

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Loss
You are selling pencils at the rate of 15 pencils for 1 rupee and incurring a loss of 9%. This means that the selling price (SP) of the pencils is less than the cost price (CP).

### Step 2: Calculate Selling Price
Since you sell 15 pencils for 1 rupee, the selling price of one pencil is:
\[ \text{SP of one pencil} = \frac{1 \text{ rupee}}{15} = \frac{1}{15} \text{ rupee} \]

### Step 3: Determine Cost Price
If you are incurring a loss of 9%, it means you are selling the pencils for 91% of their cost price. Therefore, we can express the relationship as:
\[ \text{SP} = \text{CP} \times \left(1 - \frac{9}{100}\right) = \text{CP} \times \frac{91}{100} \]

Now, substituting the SP:
\[ \frac{1}{15} = \text{CP} \times \frac{91}{100} \]

### Step 4: Solve for Cost Price
To find the cost price (CP), rearranging the equation gives:
\[ \text{CP} = \frac{1}{15} \times \frac{100}{91} = \frac{100}{1365} \text{ rupee} \]

### Step 5: Calculate New Selling Price for Gain
Now, you want to sell the pencils to gain 5%. This means the new selling price (SP') should be 105% of the cost price:
\[ \text{SP'} = \text{CP} \times \left(1 + \frac{5}{100}\right) = \text{CP} \times \frac{105}{100} \]

Substituting the value of CP:
\[ \text{SP'} = \frac{100}{1365} \times \frac{105}{100} = \frac{105}{1365} \text{ rupee} \]

### Step 6: Determine How Many Pencils for 1 Rupee
To find out how many pencils you need to sell for 1 rupee at this new selling price:
\[ \text{Number of pencils} = \frac{1 \text{ rupee}}{\text{SP'}} = \frac{1}{\frac{105}{1365}} = \frac{1365}{105} \]

Calculating this gives:
\[ \frac{1365}{105} = 13 \]

### Final Answer
Thus, you must sell **13 pencils for 1 rupee** to gain 5%.

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