Two successive trade discounts of x% and y% are allowed on the marked price of an article. Together, these two consecutive discounts are equivalent to a single discount of:

Introduction / Context:
In percentage and commercial arithmetic, discounts are often given successively, such as a first discount followed by another one on the reduced price. This question asks for a formula that combines two successive discounts of x% and y% into one equivalent discount on the original marked price.

Given Data / Assumptions:

First discount rate = x% on the marked price.
Second discount rate = y% on the already reduced price.
We seek the single equivalent discount rate D% that gives the same final selling price as applying x% and then y% consecutively.
The base (marked price) can be assumed to be any convenient value, commonly 100 units, since only proportions matter.

Concept / Approach:
Successive discounts do not simply add up, because the second discount is applied on a reduced price, not on the original price. If M is the marked price, after the first discount of x%, the price becomes M(1 - x/100). Applying a further y% discount reduces it to M(1 - x/100)(1 - y/100). An equivalent single discount D% must satisfy:
M(1 - D/100) = M(1 - x/100)(1 - y/100). Cancelling M and expanding gives a direct expression for D in terms of x and y.

Step-by-Step Solution:
Step 1: Write the final price after successive discounts. Final price = M(1 - x/100)(1 - y/100). Step 2: Expand the product. (1 - x/100)(1 - y/100) = 1 - x/100 - y/100 + xy/10000. = 1 - (x + y)/100 + xy/10000. Step 3: Represent the final price with a single discount D%. Final price = M(1 - D/100). So 1 - D/100 = 1 - (x + y)/100 + xy/10000. Step 4: Compare coefficients to find D. D/100 = (x + y)/100 - xy/10000. D = x + y - xy/100.

Verification / Alternative check:
Take a simple numerical example: x = 20%, y = 10%, M = 100. Successive discounts: after 20%, price = 80; after 10% on 80, price = 72. Single discount from 100 to 72 is 28%. Using the formula, D = 20 + 10 - (20 * 10)/100 = 30 - 2 = 28%, confirming the result.

Why Other Options Are Wrong:
(x - y + xy/100)% and (x - y - xy/100)% incorrectly involve subtraction of y, which does not match the expansion of the product (1 - x/100)(1 - y/100).
(x + y + xy/100)% overestimates the discount because it adds the interaction term xy/100 instead of subtracting it, giving a discount larger than both individual discounts combined in a realistic scenario.

Common Pitfalls:
A typical error is to simply add x% and y% and ignore the interaction term xy/100, which arises because the second discount is applied on a reduced base. Another mistake is to mis-handle the algebra when expanding the product of two binomials. Remember: for successive discounts, the effective discount is x + y - (xy/100).

Final Answer:
The equivalent single discount is (x + y - xy/100)%.