When a 20% discount is offered on a lunch buffet, the restaurant still earns a profit of 42% on the cost price. If instead the discount offered on the same marked price is 15%, then what will be the new profit percentage?

Introduction / Context:
This problem involves marked price, discount percentage and profit percentage together. The restaurant sets a marked price on the lunch buffet, gives a certain discount and still earns profit on cost price. When the discount changes, the selling price changes and so does the profit percentage. The question requires careful algebraic handling of cost price, marked price and discount to find the new profit percentage.

Given Data / Assumptions:

• Let the cost price (CP) of the lunch buffet be C.
• Let the marked price (MP) be M.
• With a 20% discount, the selling price is 0.80M and the profit is 42% of C.
• With a 15% discount, the selling price becomes 0.85M.
• We must find the new profit percentage when the discount is 15%.

Concept / Approach:
We set up the relationship between selling price and cost price using the definition of profit percentage: SP = CP * (1 + profit%). With a 20% discount, SP is 0.80M and profit is 42%, so 0.80M = 1.42C. From this, we can express M in terms of C. Then we compute the new SP at a 15% discount as 0.85M, substitute M, and finally calculate the new profit percentage using (SP − CP) / CP * 100.

Step-by-Step Solution:
Let cost price = C and marked price = M. Discount 20% means selling price SP1 = 0.80M. Given that profit at this price is 42%, so SP1 = 1.42C. Therefore, 0.80M = 1.42C. So M = (1.42 / 0.80) * C. Compute 1.42 / 0.80 = 1.775. Hence, M = 1.775C. Now, with 15% discount, new selling price SP2 = 0.85M. Substitute M: SP2 = 0.85 * 1.775C. SP2 = 1.50875C. New profit = SP2 − CP = 1.50875C − C = 0.50875C. New profit percentage = (0.50875C / C) * 100 = 50.875%.

Verification / Alternative check:
Assume a convenient cost price such as C = Rs. 100. Then, with 42% profit, SP1 = Rs. 142. Since SP1 is at 20% discount, 142 = 0.80M, so M = 142 / 0.80 = Rs. 177.5. With a 15% discount, SP2 = 0.85 * 177.5 = Rs. 150.875. Profit now is 150.875 − 100 = Rs. 50.875, which as a percentage of 100 is 50.875%. This confirms our algebraic result.

Why Other Options Are Wrong:
57% and 63.125%: These correspond to incorrect manipulations of discount and profit rates or treating percentage changes additively instead of using the correct relationships.
44.75% and 48%: These are plausible but arise from taking intermediate rounding or misinterpreting the initial 42% profit relation.
Only 50.875% is consistent with both the algebraic equation and a numeric check with an assumed cost price.

Common Pitfalls:
Students sometimes directly adjust the profit percentage by the change in discount (20% to 15%) and subtract 5% from 42%, which is incorrect. Others mix up the roles of CP and MP or forget that discount is applied on the marked price, not on the cost price. Always write down the equations for SP in terms of CP and MP, use one scenario to link M and C, and then evaluate the second scenario carefully.

Final Answer:
When the discount is 15%, the profit percentage becomes approximately 50.875%.