By selling 15 pencils for Rs.30, Rajan lost 20%. How many pencils should he sell for Rs.52 to have
a gain of 30%?

To solve this problem, let's break it down step by step.

### Step 1: Find the Cost Price (CP) per pencil.
Rajan sold 15 pencils for Rs. 30 at a 20% loss. 

Let the Cost Price of 15 pencils be \( CP_{15} \).

Given the loss is 20%, the Selling Price (SP) is:
\[
SP = CP_{15} - \text{Loss}
\]
Since the loss is 20% of \( CP_{15} \):
\[
SP = CP_{15} - 0.2 \times CP_{15} = 0.8 \times CP_{15}
\]
We are given that the SP is Rs. 30, so:
\[
0.8 \times CP_{15} = 30
\]
\[
CP_{15} = \frac{30}{0.8} = 37.5 \text{ Rs.}
\]
Thus, the cost price of 15 pencils is Rs. 37.5.

The cost price per pencil \( CP \) is:
\[
CP = \frac{CP_{15}}{15} = \frac{37.5}{15} = 2.5 \text{ Rs.}
\]

### Step 2: Determine the number of pencils to sell for Rs. 52 to gain 30%.
Let the number of pencils Rajan needs to sell for Rs. 52 be \( x \).

For a 30% gain, the Selling Price per pencil \( SP \) should be:
\[
SP = CP \times (1 + \text{Gain\%})
\]
\[
SP = 2.5 \times (1 + 0.3) = 2.5 \times 1.3 = 3.25 \text{ Rs. per pencil}
\]

### Step 3: Find the number of pencils.
The total Selling Price for \( x \) pencils is Rs. 52:
\[
3.25 \times x = 52
\]
\[
x = \frac{52}{3.25} = 16
\]

Thus, Rajan needs to sell **16 pencils** for Rs. 52 to gain 30%.

Would you like any more details or have any other questions?

### Related Questions:
1. How do you calculate the selling price given a certain percentage loss?
2. What would the cost price be if Rajan sold 10 pencils at a 10% profit?
3. How many pencils should Rajan sell to break even if the cost price per pencil is Rs. 2.5?
4. If Rajan wants to gain 50% instead of 30%, how many pencils should he sell for Rs. 52?
5. What is the formula to calculate the percentage gain or loss in a transaction?
6. How would the selling price change if Rajan decided to sell the pencils at a 10% loss instead?
7. Can you derive the formula used to determine the number of items needed to achieve a certain profit margin?
8. If Rajan sold 20 pencils at a 15% gain, what would be the total selling price?

### Tip:
Always check whether the percentage given (gain or loss) applies to the cost price or selling price to avoid mistakes.

Learn how to calculate the number of pencils Rajan should sell for Rs. 52 to achieve a 30% gain. Step-by-step solution includes finding cost price per pencil and applying gain percentage formula. Suitable for Grades 7-9.

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