(a % of a) + (b % of b) = 2% of ab, then what percentage of a is b ?

Concept

For any value v, "x% of v" equals (x/100) × v. When two quantities a and b satisfy a symmetric percentage equation, expand every term algebraically and simplify — if the result reduces to a perfect-square identity (p − q)² = 0, then p and q must be equal, so one quantity is exactly 100% of the other. (As is standard in such aptitude problems, a and b are taken to be positive real numbers.)

Step-by-step solution

1. Expand each percentage term: a% of a = (a/100)·a = a2/100, and b% of b = (b/100)·b = b2/100.

2. Expand the right-hand side: 2% of ab = (2/100)·ab = 2ab/100.

3. Set up the equation from the given relation: a2/100 + b2/100 = 2ab/100.

4. Multiply every term by 100 to clear the denominators: a2 + b2 = 2ab.

5. Bring all terms to one side: a2 − 2ab + b2 = 0.

6. Recognize the left side as a perfect square: (a − b)2 = 0.

7. Take the square root of both sides: a − b = 0, so a = b.

8. Since a equals b, b is exactly 100% of a.

Cross-check

Substitute a = b = k back into the original equation: (k/100)·k + (k/100)·k = 2k2/100, which equals 2% of k·k = 2% of ab — matching the given condition exactly and confirming the result.

Answer

Therefore, b is 100% of a.