A dealer marks his goods at 20% above the cost price and then allows a discount so that he finally makes a profit of 8%. What is the rate of discount he must offer on the marked price?

Introduction / Context:
This question connects cost price, marked price, discount and profit percentage. It represents a typical shopkeeper scenario where articles are first marked up from cost and then sold after giving some discount. To solve such problems, students must clearly understand the relation between these quantities and form equations using percentage factors. The aim here is to find the discount rate that converts a 20% markup into an effective profit of 8% after discount.

Given Data / Assumptions:

• Let the cost price (CP) of the article be C rupees.
• Marked price (MP) is 20% above CP, so MP = C * (1 + 20/100) = 1.2C.
• After allowing some discount on MP, the article is sold at a selling price (SP) that yields an 8% profit on CP.
• Thus SP = C * (1 + 8/100) = 1.08C.
• If discount rate is d%, then SP = MP * (1 − d/100).

Concept / Approach:
We equate the two expressions for SP. On one side, SP equals 1.08C from the profit condition. On the other side, SP equals 1.2C * (1 − d/100) from the discount condition. By solving this equation for d, we obtain the required discount rate. The method uses the idea that the cost price is the base for profit, while the marked price is the base for discount, and these two are connected through the relationships described above.

Step-by-Step Solution:
Step 1: Assume cost price = C. Step 2: Marked price = C * (1 + 20/100) = 1.2C. Step 3: Desired selling price for 8% profit = C * (1 + 8/100) = 1.08C. Step 4: Let discount rate on marked price be d%. Then SP = MP * (1 − d/100) = 1.2C * (1 − d/100). Step 5: Equate SP expressions: 1.2C * (1 − d/100) = 1.08C. Step 6: Cancel C: 1.2 * (1 − d/100) = 1.08. So (1 − d/100) = 1.08 / 1.2 = 0.9. Step 7: Therefore 1 − d/100 = 0.9 implies d/100 = 0.1, so d = 10%.

Verification / Alternative check:
Take C = Rs 100 for simplicity. Then MP = 1.2 * 100 = Rs 120. Discount of 10% on Rs 120 is Rs 12, so SP = 120 − 12 = Rs 108. Profit = SP − CP = 108 − 100 = Rs 8, which is 8% of 100. This matches the required profit percentage and confirms that a 10% discount on the marked price is correct.

Why Other Options Are Wrong:

• 4% discount would give SP = 1.2C * 0.96 = 1.152C, which is a 15.2% profit, not 8%.
• 6% discount leads to SP = 1.2C * 0.94 = 1.128C, equal to 12.8% profit.
• 12% discount gives SP = 1.2C * 0.88 = 1.056C, which is only 5.6% profit.

Common Pitfalls:
Students often mistakenly apply the discount to the cost price instead of the marked price. Another error is to mix up profit percent and discount percent bases and try to add or subtract them directly. Some also assume that if markup is 20% and net profit is 8%, then discount is simply 12%, which ignores compounding via multiplication of factors. Always base calculations on clear algebraic equations relating CP, MP, discount and SP.

Final Answer:
The discount rate that the dealer must offer is 10%.