The ratio of the cost prices of articles X and Y is 2 : 9. A man earns a…

When two articles with cost prices in a given ratio are each sold at a different gain or loss percentage, the overall gain/loss percentage on the combined transaction is never the simple average of the two individual percentages. It must be found from the TOTAL cost price and TOTAL selling price of both articles together, weighting each individual percentage by that article's own share of the combined cost.

1. Using the given ratio 2 : 9, let the cost price of X be 2k and the cost price of Y be 9k (k a common unit).

2. Selling price of X, after a 31% gain = 2k × (100 + 31)/100 = 2k × 1.31 = 2.62k.

3. Selling price of Y, after a 13% loss = 9k × (100 − 13)/100 = 9k × 0.87 = 7.83k.

4. Total cost price of both articles = 2k + 9k = 11k.

5. Total selling price of both articles = 2.62k + 7.83k = 10.45k.

6. Since the total selling price (10.45k) is less than the total cost price (11k), the transaction results in an overall loss of 11k − 10.45k = 0.55k.

7. Overall loss percentage = (0.55k / 11k) × 100 = 5%.

Cross-check with the weighted-average shortcut: weight each individual percentage by its article's share of the combined 11 parts — Overall % = (2 × (+31) + 9 × (−13)) / (2 + 9) = (62 − 117) / 11 = −55/11 = −5%. The negative sign denotes a loss, matching the 5% loss found above by the total-cost/total-sale method.

The entire transaction results in a loss of 5%.