If y exceeds x by 15%, then x is less than y by what percent?

Concept — A percentage change is always measured against the base quantity that comes after the word "than". So "A is less than B by what percent" uses B as the base: percent = (B − A) ÷ B × 100. Because the base differs between the two comparisons, "y more than x" and "x less than y" are not equal percentages.

Application

1. y exceeds x by 15%, so y = x + 15% of x = 1.15x (here the base is x).

2. We now need how far x falls short of y, so the base switches to the larger quantity y.

3. Difference = y − x = 1.15x − x = 0.15x.

4. Required percent = (y − x) ÷ y × 100 = 0.15x ÷ 1.15x × 100 = 15/115 × 100 = 3/23 × 100 = 300/23 %.

5. 300/23 = 13 1/23 % (since 23 × 13 = 299, leaving a remainder of 1).

Cross-check — If x is 3/23 short of y, then x = (20/23)y. Working backwards, y is greater than x by (3/23 y) ÷ (20/23 y) = 3/20 = 15%, which matches the given data — so 13 1/23% is consistent.