A man purchased a box full of pencils at the rate of 7 for Rs. 9 and sold all of them at the rate of 8 for Rs. 12. In this transaction, he gained Rs. 10. How many pencils did did the box contian?

Answer

To solve the problem step by step, we need to determine how many pencils were in the box based on the given cost price (CP) and selling price (SP) and the profit made.

### Step-by-Step Solution:

1. **Determine the Cost Price (CP) and Selling Price (SP) per pencil:**
- The cost price for 7 pencils is Rs. 9.
- Therefore, the cost price per pencil (CP) = \( \frac{9}{7} \) Rs.

- The selling price for 8 pencils is Rs. 12.
- Therefore, the selling price per pencil (SP) = \( \frac{12}{8} = \frac{3}{2} \) Rs.

2. **Calculate the profit per pencil:**
- Profit per pencil = SP - CP
- Profit per pencil = \( \frac{3}{2} - \frac{9}{7} \)

To subtract these fractions, we need a common denominator. The least common multiple of 2 and 7 is 14.

- Convert \( \frac{3}{2} \) to a fraction with a denominator of 14:
\( \frac{3}{2} = \frac{3 \times 7}{2 \times 7} = \frac{21}{14} \)

- Convert \( \frac{9}{7} \) to a fraction with a denominator of 14:
\( \frac{9}{7} = \frac{9 \times 2}{7 \times 2} = \frac{18}{14} \)

- Now, subtract:
Profit per pencil = \( \frac{21}{14} - \frac{18}{14} = \frac{3}{14} \) Rs.

3. **Determine the total profit:**
- We know the total profit from selling all the pencils is Rs. 10.
- Let the number of pencils in the box be \( x \).
- Therefore, the total profit can also be expressed as:
Total Profit = Profit per pencil × Number of pencils
\( 10 = \frac{3}{14} \times x \)

4. **Solve for \( x \):**
- Rearranging the equation gives:
\( x = 10 \times \frac{14}{3} \)
\( x = \frac{140}{3} \)

- Calculating this gives:
\( x = 46.67 \)

Since the number of pencils must be a whole number, we round it to the nearest whole number. Thus, the box contains approximately 46 pencils.

### Final Answer:
The box contains **46 pencils**.