The ratio of the cost prices of articles X and Y is 3:7. A man earns a profit…

When two items are sold together at cost prices in a given ratio, but each earns a different percentage profit or loss, the combined profit/loss percentage on the whole transaction is found from the total selling price versus the total cost price -- not by averaging the two percentages directly.

1. Let the cost prices of X and Y be in the ratio 3:7, so take CP(X) = 3k and CP(Y) = 7k for some positive number k.

2. A profit of 46% on X gives SP(X) = 3k x (1 + 46/100) = 3k x 1.46 = 4.38k.

3. A loss of 39% on Y gives SP(Y) = 7k x (1 - 39/100) = 7k x 0.61 = 4.27k.

4. Total cost price = CP(X) + CP(Y) = 3k + 7k = 10k.

5. Total selling price = SP(X) + SP(Y) = 4.38k + 4.27k = 8.65k.

6. Since the total selling price (8.65k) is less than the total cost price (10k), the transaction results in a loss = 10k - 8.65k = 1.35k.

7. Loss percentage = (Loss / Total CP) x 100 = (1.35k / 10k) x 100 = 13.5%.

Cross-check with the weighted-ratio shortcut: overall percentage = (3 x 46 + 7 x (-39)) / (3 + 7) = (138 - 273) / 10 = -13.5%, where the negative sign denotes a loss -- confirming the step-by-step result.

The entire transaction therefore results in a loss of 13.5%.