The selling price of a mat is five times the discount offered on it. If the percentage discount is equal to the percentage profit, what is the ratio of the discount offered to the cost price of the mat?

Introduction / Context:
This is an algebraic profit and discount question involving a relationship between selling price, discount and profit percentages. The condition that the selling price is a multiple of the discount amount, together with the equality of discount percentage and profit percentage, creates equations that link the cost price, marked price and discount. The task is to find the ratio of the discount amount to the cost price of the mat, which is a common type of ratio based question in aptitude exams.

Given Data / Assumptions:

• Selling price (SP) is five times the discount (D), so SP = 5D.
• Let marked price (MP) be M.
• Discount amount D = M - SP.
• Discount percentage = (D / M) * 100.
• Profit percentage = (SP - CP) / CP * 100, where CP is cost price.
• Given that discount percentage equals profit percentage.
• We must find the ratio of discount amount D to cost price CP, that is D : CP.

Concept / Approach:
We first express SP in terms of D and then relate M to D using the definition of discount amount. From SP = 5D and D = M - SP, we get M = 6D. Discount percentage becomes (D / 6D) * 100 = (1/6) * 100. Since profit percentage must equal this value, we equate (SP - CP) / CP to 1/6 in fractional form. With SP expressed as 5D, we obtain a linear equation linking D and CP. Simplifying that equation gives the desired ratio D : CP.

Step-by-Step Solution:
Step 1: Let discount amount be D and marked price be M. Step 2: Given SP = 5D. Step 3: By definition of discount, D = M - SP = M - 5D, so M = 5D + D = 6D. Step 4: Discount percentage = (D / M) * 100 = (D / 6D) * 100 = (1/6) * 100. Step 5: So discount percent = profit percent = 1/6 in fractional form. Step 6: Let cost price be CP. Step 7: Profit percent = (SP - CP) / CP = 1/6. Step 8: Substitute SP = 5D: (5D - CP) / CP = 1/6. Step 9: Cross multiply: 6(5D - CP) = CP. Step 10: Expand: 30D - 6CP = CP. Step 11: Rearrange: 30D = 7CP → CP = (30/7)D. Step 12: Therefore, D : CP = D : (30/7 * D) = 7 : 30.

Verification / Alternative check:
Take a convenient discount value, for example D = Rs. 7. Then SP = 5D = Rs. 35 and M = 6D = Rs. 42. Discount percentage = (7 / 42) * 100 = 1/6 * 100 ≈ 16.67%. From CP = (30/7) * 7 = Rs. 30, profit = SP - CP = 35 - 30 = Rs. 5. Profit percent = (5 / 30) * 100 = 1/6 * 100 ≈ 16.67%. Thus both discount and profit percentages match, confirming that D : CP = 7 : 30 is correct.

Why Other Options Are Wrong:
Ratios such as 6:31, 11:30, 31:6 or 5:24 do not satisfy the simultaneous conditions SP = 5D and equal discount and profit percentages. Substituting any of these ratios into the equations would break either the relationship between SP and D or the equality of discount percent and profit percent.

Common Pitfalls:
A common mistake is to confuse discount percentage with discount amount or to treat SP as multiple of the percentage rather than of the amount. Another error is to set profit percentage equal to discount amount fraction without converting both to proper fractional forms. Writing all relationships algebraically and keeping track of which quantity is which prevents these errors.

Final Answer:
The required ratio of discount to cost price is 7:30.