Question

If a% of a + b% of b = 2% of ab,then what percent of a is b?

• A. 25%
• B. 50%
• C. 75%
• D. 100%

Answer 1

The Correct Option is D

To solve the problem, we need to find what percent of \(a\) is \(b\) given the equation:

\(a\%\ of\ a\ +\ b\%\ of\ b = 2\%\ of\ ab\)

Let's break down the expression:

1. Convert the percentages into their fractional forms: \(\frac{a}{100} \times a + \frac{b}{100} \times b = \frac{2}{100} \times ab\)
2. Simplify the terms: \(\frac{a^2}{100} + \frac{b^2}{100} = \frac{2ab}{100}\)
3. Multiply through by 100 to eliminate fractions: \(a^2 + b^2 = 2ab\)
4. Rearrange to form a quadratic equation: \(a^2 - 2ab + b^2 = 0\)
5. Recognize the left side as a perfect square: \((a - b)^2 = 0\)
6. Take the square root of both sides: \(a - b = 0\)
7. Solve for \(b\): \(b = a\)

Since \(b = a\), \(b\) is 100% of \(a\).

Thus, the correct answer is 100%.

Answer 2

The given equation is:

\[ \frac{a}{100} \times a + \frac{b}{100} \times b = \frac{2}{100} \times ab \]

Simplifying this equation step-by-step:

\[ \frac{a^2}{100} + \frac{b^2}{100} = \frac{2ab}{100} \]

Multiply both sides by 100 to eliminate the denominator:

\[ a^2 + b^2 = 2ab \]

This simplifies to:

\[ a^2 - 2ab + b^2 = 0 \]

Factoring the equation:

\[ (a - b)^2 = 0 \]

From this, we get:

\[ a = b \]

Thus, \( b \) is equal to \( a \), so \( b \) is 100% of \( a \).