If y exceeds x by 30% then x is less than y by what percent?

Concept

When a quantity y is larger than a quantity x by R%, the percent by which x falls short of y is NOT the same R%, because the base of comparison changes from x to y. The reverse percentage is R/(100 + R) × 100.

Application

1. Let x be the smaller quantity. y exceeds x by 30%, so y = x + 30% of x = x(1 + 30/100) = 1.3x.

2. The amount by which x falls short of y is y − x = 1.3x − x = 0.3x.

3. x is less than y by (y − x)/y × 100 = 0.3x/1.3x × 100. The x cancels, leaving 0.3/1.3 × 100 = 30/130 × 100.

4. Simplify 30/130 = 3/13, so the answer = 3/13 × 100 = 300/13 % = 23 1/13 %.

Cross-check

Using the direct identity R/(100 + R) × 100 with R = 30: 30/130 × 100 = 300/13 = 23 1/13 %, which matches the step-by-step result.