Question

If the numerator of a rational number is decreased by 20% and the denominator is increased by 150%, the resultant number becomes \( \frac{1}{2} \). What is the original number?

• A. \( \frac{5}{4} \)
• B. \( \frac{4}{5} \)
• C. \( \frac{16}{25} \)
• D. \( \frac{25}{16} \)

Hint:

When solving problems involving percentages in ratios, express the changes as multiplication factors for easier computation.

Answer 1

The Correct Option is D

Let the original rational number be \( \frac{a}{b} \). The numerator is decreased by 20%, so the new numerator is \( 0.80a \). The denominator is increased by 150%, so the new denominator is \( 2.5b \). We are given that the resultant number becomes \( \frac{1}{2} \), so: \[ \frac{0.80a}{2.5b} = \frac{1}{2} \] Cross-multiply to solve for \( a \) and \( b \): \[ 2 \times 0.80a = 2.5b \] \[ 1.60a = 2.5b \] \[ \frac{a}{b} = \frac{2.5}{1.60} = \frac{25}{16} \] Thus, the original rational number is \( \frac{25}{16} \).