If 35% of (x + y) is equal to 40% of (x − y), then x is what percentage of y?

Introduction / Context:

This is an algebraic percentage problem involving two variables x and y. You are given an equation where percentages of sums and differences of x and y are equal, and you need to express x as a percentage of y. This type of problem mixes percentages with basic algebra and is often seen in algebra and percentage sections of aptitude tests.

Given Data / Assumptions:

• 35% of (x + y) equals 40% of (x − y).
• The variables x and y are real numbers and y is nonzero so that the percentage comparison makes sense.
• We must find how many percent x is of y.

Concept / Approach:

First, we convert the percentage statement into an equation using decimal or fractional equivalents. Then, we simplify the equation to get a relationship between x and y. Once we express x in terms of y, we convert the ratio x / y into a percentage by multiplying by 100. The key is to carefully distribute and combine like terms.

Step-by-Step Solution:

Verification / Alternative check:

To verify, choose a convenient value for y, for example y = 1. Then x = 15 * 1 = 15. Now compute both sides of the original statement. Left side: 35% of (x + y) = 35% of (15 + 1) = 35% of 16 = 0.35 * 16 = 5.6. Right side: 40% of (x − y) = 40% of (15 − 1) = 40% of 14 = 0.40 * 14 = 5.6. Both sides are equal, confirming the relationship.

Why Other Options Are Wrong:

• 150% and 200%: These would imply that x is only slightly larger than y, which contradicts the derived equation x = 15y.
• 105%: This is just a little more than y and clearly does not satisfy the algebraic relationship when tested.
• 15%: This would imply x is smaller than y, again inconsistent with x being 15 times y.

Common Pitfalls:

Common mistakes include cancelling incorrectly, forgetting to distribute the percentages across both x and y terms, or directly equating coefficients without expanding properly. Another error is misinterpreting 15 as 15% instead of 1500%. Always remember that if x = ky, then x as a percentage of y is k * 100%, not just k.

Final Answer:

The value of x is 1500% of y.