Mixture of equal counts at different buying rates: A man buys an equal number of toffees at 6 per rupee and at 7 per rupee. He mixes all the toffees and sells the entire lot at 6 per rupee. Find his overall gain or loss percentage on the transaction.

Introduction / Context:
Weighted-average cost problems often give buying rates as “k per rupee.” Converting these to per-toffee costs and working with counts makes the effective average cost clear. The selling rate is common, so revenue is easy to compute.

Given Data / Assumptions:

• Equal number of toffees purchased at 6 per rupee and 7 per rupee.
• All toffees sold at 6 per rupee.
• Let each batch contain n toffees for easy algebra.

Concept / Approach:
Cost per toffee when 6 per rupee is 1/6 rupee; when 7 per rupee is 1/7 rupee. Total cost is the sum across equal counts. Total revenue uses the selling price 1/6 rupee per toffee times total pieces. Profit% = (profit / cost) * 100.

Step-by-Step Solution:
Cost for first n toffees = n * (1/6) rupeeCost for second n toffees = n * (1/7) rupeeTotal cost = n(1/6 + 1/7) = n * (13/42) rupeeTotal toffees = 2n; Selling rate = 1/6 rupee each ⇒ Total revenue = 2n * (1/6) = n/3 rupeeProfit = revenue − cost = n/3 − n(13/42) = n(14/42 − 13/42) = n/42 rupeeProfit% = ( (n/42) / (n * 13/42) ) * 100 = (1/13) * 100 = 100/13 % ≈ 7.692%

Verification / Alternative check:
Choose n = 42 for integers: Cost = 42(13/42) = 13 rupees; Revenue = (2*42)/6 = 14 rupees; Profit = 1 rupee ⇒ Profit% = 1/13 * 100 = 100/13 %.

Why Other Options Are Wrong:

• 69/13 % loss / 76/13 % loss: sign is wrong; the computed outcome is a gain.
• 79/13 % gain: lower than the exact 100/13 %; arises from arithmetic slips.

Common Pitfalls:

• Averaging the rupee rates instead of computing per-toffee costs and total revenue.
• Using the wrong base (selling price instead of cost) for the percentage.

Final Answer:
100/13 % gain