When a sweater is sold with a 20% discount on its marked price, the shopkeeper still makes a profit of 28%. If the discount is reduced to 14% on the same marked price, what will be the new profit percentage?

Introduction / Context:
This is another profit and loss question involving a change in discount while the marked price remains the same. The idea is to deduce the cost price from one scenario where both discount and profit are known, and then use that cost price to compute profit in a second scenario with a different discount. These questions sharpen your understanding of how discount percentages on marked price relate to profit percentages on cost price.

Given Data / Assumptions:

• Let the marked (list) price be M.
• First case: discount = 20%, so SP1 = 0.80M.
• Under SP1, profit = 28%, so SP1 = 1.28 * CP.
• Second case: discount = 14%, so SP2 = 0.86M.
• We need the new profit percentage in the second case.
• All other factors like taxes and costs are assumed unchanged.

Concept / Approach:
Method:

• Use 0.80M = 1.28 * CP to find CP in terms of M.
• Then compare SP2 = 0.86M with CP to compute profit percentage.
• Profit percentage = ((SP2 - CP) / CP) * 100.

The key is that CP is fixed while the marked price and discount determine different selling prices.

Step-by-Step Solution:
Step 1: From first case, SP1 = 0.80M. Step 2: Profit of 28% implies SP1 = 1.28 * CP. Step 3: So 0.80M = 1.28 * CP, hence CP = 0.80M / 1.28. Step 4: Simplify CP: CP = (80 / 128)M = (5 / 8)M = 0.625M. Step 5: In second case, SP2 = 0.86M (14% discount). Step 6: Profit fraction = (SP2 - CP) / CP = (0.86M - 0.625M) / 0.625M. Step 7: Numerator = 0.235M, denominator = 0.625M. Step 8: Profit fraction = 0.235 / 0.625 = 0.376. Step 9: Profit percentage = 0.376 * 100 = 37.6%.

Verification / Alternative check:
Take an easy value for the marked price, say M = Rs 1000:

• First case: SP1 = 0.80 * 1000 = Rs 800.
• Profit 28% means CP = SP1 / 1.28 = 800 / 1.28 = Rs 625.
• Second case: SP2 = 0.86 * 1000 = Rs 860.
• Profit = SP2 - CP = 860 - 625 = Rs 235.
• Profit percentage = 235 / 625 * 100 = 37.6%.

The numerical check confirms the algebraic calculation and removes any ambiguity.

Why Other Options Are Wrong:
42 percent, 46.4 percent, and 33.2 percent all come from incorrect handling of the relationship between discount and profit. They might arise if someone mistakenly treats profit percentages directly in relation to the marked price or confuses SP1 and SP2 in calculations. None of these values exactly match the correct ratio between SP2 and CP, which we have verified carefully.

Common Pitfalls:
A common mistake is assuming that reducing the discount by a certain percentage automatically increases profit by the same percentage, which is incorrect. Another error is treating the 28% profit as being on the marked price rather than on cost price. You must always be clear about which base each percentage is applied to: discount on marked price, profit or loss on cost price. Keeping this distinction straight is essential for accuracy.

Final Answer:
With a 14% discount on the marked price, the shopkeeper earns a 37.6 percent profit on the sweater.