A man bought pencils at the rate of 6 for ₹ 4 and sold them at the rate of 4 for ₹ 6. His gain in the transaction is

### Concept & Formula The gain percent in a transaction where items are bought and sold at different rates can be calculated by equalizing the number of items or finding the per-item cost. $$Gain \% = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100$$ ### Step-by-Step Solution * **Given:** CP rate = 6 pencils for ₹ 4. SP rate = 4 pencils for ₹ 6. * **Calculation:** 1. CP of 1 pencil = $\frac{4}{6} = \frac{2}{3}$ 2. SP of 1 pencil = $\frac{6}{4} = \frac{3}{2}$ 3. Gain = $SP - CP = \frac{3}{2} - \frac{2}{3} = \frac{9 - 4}{6} = \frac{5}{6}$ 4. Gain \% = $\left( \frac{5/6}{2/3} \right) \times 100 = \left( \frac{5}{6} \times \frac{3}{2} \right) \times 100$ 5. Gain \% = $\frac{15}{12} \times 100 = \frac{5}{4} \times 100 = 125\%$ ### Exam Strategy & Shortcut LCM Method: LCM of quantities (6 and 4) is 12. Assume 12 pencils were bought and sold. Total CP for 12 pencils = $4 \times 2 = 8$ Total SP for 12 pencils = $6 \times 3 = 18$ Gain = $18 - 8 = 10$ Gain \% = $\left(\frac{10}{8}\right) \times 100 = 125\%$. ### Common Pitfall Miscalculating the fractions when subtracting the Cost Price from the Selling Price. Utilizing the LCM method entirely bypasses fractional subtraction, minimizing errors. ### Final Answer Therefore, the correct answer is **125%**.