Two successive discounts of x% and y% on the marked price of an article are equivalent to a single discount of what percentage on the marked price?

Introduction / Context:This is a conceptual question about successive discounts in percentage form. In real life, shops may offer two consecutive discounts on a marked price, and it is often useful to know what single equivalent discount these two reductions represent. The key point is that successive discounts do not simply add; instead they interact multiplicatively, leading to an effective discount that is slightly less than the arithmetic sum of the individual discounts.

Given Data / Assumptions:

• First discount = x% on the marked price.
• Second discount = y% applied on the reduced price after the first discount.
• Both x and y are expressed as percentages.
• We are asked to find the single equivalent discount percentage on the original marked price.

Concept / Approach:Successive discounts are handled by multiplying the remaining fractions of price rather than simply adding discount percentages. If the marked price is M, then:

• After x% discount, price becomes M * (1 − x/100).
• After a further y% discount, price becomes M * (1 − x/100) * (1 − y/100).

The equivalent single discount D% must satisfy: final price = M * (1 − D/100). We equate the two expressions and solve for D, which leads to the standard formula D = x + y − (xy/100).

Step-by-Step Solution:Step 1: Let marked price be M.Step 2: After first discount of x%, price = M * (1 − x/100).Step 3: After second discount of y%, price = M * (1 − x/100) * (1 − y/100).Step 4: Expand (1 − x/100) * (1 − y/100) = 1 − x/100 − y/100 + xy/10000.Step 5: Write it as 1 − (x + y)/100 + xy/10000.Step 6: Suppose equivalent single discount is D%. Then final price should also equal M * (1 − D/100).Step 7: Therefore 1 − D/100 = 1 − (x + y)/100 + xy/10000.Step 8: Comparing terms, D/100 = (x + y)/100 − xy/10000.Step 9: Multiply by 100: D = x + y − xy/100.

Verification / Alternative check:Take a numerical example: let x = 20% and y = 10%. Then equivalent discount should be D = 20 + 10 − (20 * 10)/100 = 30 − 2 = 28%. If M = Rs. 100, two successive discounts give price = 100 * 0.8 * 0.9 = 72, a discount of Rs. 28, or 28%. This confirms the formula and the chosen option.

Why Other Options Are Wrong:(x − y + xy/100)% and (x − y − xy/100)% incorrectly subtract y% or mis-handle the interaction term. (x + y + xy/100)% makes the equivalent discount even larger than x + y, which is impossible because successive discounts are always slightly less than their sum. The expression (x + y − 2xy/100)% overcompensates for the interaction term and is not derived from the correct algebraic expansion.

Common Pitfalls:

• Simply adding x and y and ignoring the multiplicative nature of successive discounts.
• Confusing successive discounts with successive percentage increases.
• Making algebraic errors in expanding (1 − x/100)(1 − y/100).
• Assuming the interaction term xy/100 is negligible for large x and y, which can create sizable errors.

Final Answer:The single equivalent discount is (x + y − xy/100)% on the marked price.