If x% of a is the same as y% of b, then z% of b is :

Concept: When two quantities are linked by an equality of percentages, such as x% of a = y% of b, clearing the percentage denominators turns it into a linear relationship (xa = yb) that lets you express one variable in terms of the other. Substituting that expression into any other percentage of the same variable converts it into an equivalent percentage of the other variable.

Application

1. Write the given equality as fractions: x% of a = y% of b means (x/100)·a = (y/100)·b.

2. Multiply both sides by 100 to clear denominators: xa = yb.

3. Solve for b: b = xa/y.

4. Write z% of b using this substitution: z% of b = (z/100)·(xa/y).

5. Regroup the factors so a/100 is isolated: (z/100)·(xa/y) = (xz/y)·(a/100), which is exactly (xz/y)% of a.

Cross-check: Take x = 10, a = 100, so x% of a = 10. Let y = 20; then y% of b = 10 gives b = 50, matching b = xa/y = (10·100)/20 = 50. For z = 5, z% of b = 5% of 50 = 2.5, and (xz/y)% of a = (10·5/20)% of 100 = 2.5% of 100 = 2.5 — the two values agree.

Result: z% of b simplifies to (xz/y)% of a.