A man bought apples at the rate of 8 for ₹ 34 and sold them at the rate of 12 for ₹ 57. How many apples should be sold to earn a net profit of ₹ 45?

### Concept & Unitary Method When dealing with bulk buying and selling rates, the most straightforward approach is to calculate the Cost Price (CP) and Selling Price (SP) of a single item. The profit on one item can then be scaled up to meet the total required profit. $$ \text{Total Required Quantity} = \frac{\text{Total Target Profit}}{\text{Profit per Item}} $$ ### Step-by-Step Solution 1. **Find CP of 1 apple:** He buys 8 apples for ₹ 34. $$ \text{CP of 1 apple} = \frac{34}{8} = \text{₹ } 4.25 $$ 2. **Find SP of 1 apple:** He sells 12 apples for ₹ 57. $$ \text{SP of 1 apple} = \frac{57}{12} = \text{₹ } 4.75 $$ 3. **Find Profit on 1 apple:** $$ \text{Profit on 1 apple} = \text{SP} - \text{CP} = 4.75 - 4.25 = \text{₹ } 0.50 $$ 4. **Find the number of apples needed for ₹ 45 profit:** $$ \text{Number of apples} = \frac{\text{Total Target Profit}}{\text{Profit on 1 apple}} $$ $$ \text{Number of apples} = \frac{45}{0.50} $$ $$ \text{Number of apples} = 90 $$ ### Exam Strategy & Shortcut Instead of decimals, use LCM to equate the number of apples bought and sold. LCM of 8 and 12 is 24. Buy 24 apples: $34 \times 3 = \text{₹ } 102$. Sell 24 apples: $57 \times 2 = \text{₹ } 114$. Profit on 24 apples = $114 - 102 = \text{₹ } 12$. To get ₹ 12 profit, he sells 24 apples. To get ₹ 45 profit, he sells: $(24 / 12) \times 45 = 2 \times 45 = 90$ apples. ### Common Pitfall Getting lost in decimal calculations ($34/8$ or $57/12$) can lead to arithmetic errors. If the decimals look messy, immediately switch to the LCM method to keep numbers whole. ### Final Answer Therefore, the correct answer is **90**.