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, we need to determine the cost price of one pencil and then use it to find out how many pencils Rajan should sell to achieve a 30% gain.

### Step 1: Determine the cost price of 15 pencils
Rajan sold 15 pencils for Rs. 30 at a loss of 20%. This means that Rs. 30 is 80% of the actual cost price of the 15 pencils.

Let the cost price of 15 pencils be \( C \).

Since he incurred a loss of 20%, we have:
\[
0.8C = 30
\]

To find \( C \), divide both sides by 0.8:
\[
C = \frac{30}{0.8} = 37.5
\]

So, the cost price of 15 pencils is Rs. 37.5.

### Step 2: Determine the cost price of one pencil
The cost price of one pencil is:
\[
\text{Cost price of 1 pencil} = \frac{37.5}{15} = 2.5 \text{ Rs.}
\]

### Step 3: Find the selling price for a 30% gain
For a 30% gain, the selling price of each pencil should be:
\[
\text{Selling price of 1 pencil} = 2.5 \times 1.3 = 3.25 \text{ Rs.}
\]

### Step 4: Determine the number of pencils to sell for Rs. 52
Let \( x \) be the number of pencils Rajan needs to sell to get Rs. 52 at a 30% gain.

\[
3.25x = 52
\]

Solve for \( x \):
\[
x = \frac{52}{3.25} = 16
\]

So, Rajan should sell 16 pencils for Rs. 52 to have a gain of 30%.

Would you like any more details or have any questions?

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Here are 8 related questions for further exploration:

1. What would be the selling price if Rajan wanted to break even (no gain, no loss)?
2. How much would Rajan gain if he sold 20 pencils at the price calculated for a 30% gain?
3. What would be the percentage gain if Rajan sold 10 pencils for Rs. 35?
4. If the cost price of pencils increased by 10%, how many pencils should Rajan sell for Rs. 60 to gain 20%?
5. How much loss would Rajan incur if he sold 12 pencils for Rs. 24?
6. What is the total profit when Rajan sells 25 pencils at a 30% gain?
7. How many pencils should Rajan sell for Rs. 45 to achieve a 15% profit?
8. What would be the loss percentage if Rajan sold 15 pencils for Rs. 25?

**Tip:** In percentage gain or loss problems, understanding the relationship between cost price, selling price, and the percentage is crucial for solving the problem correctly.

Learn how to calculate the number of pencils Rajan should sell for Rs. 52 to achieve a 30% gain. This problem involves determining cost price, calculating selling price for a gain, and applying percentage change formulas. Suitable for high school students.

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