Bonus Problem. By what percent will the fraction change if its numerator is increased by 25% and its denominator is increased by

20%?

1 answer

Let the original fraction be represented as \( \frac{a}{b} \).

When the numerator is increased by 25%, the new numerator becomes: \[ a + 0.25a = 1.25a. \]

When the denominator is increased by 20%, the new denominator becomes: \[ b + 0.20b = 1.20b. \]

Thus, the new fraction is: \[ \frac{1.25a}{1.20b}. \]

Now, we can simplify the new fraction: \[ \frac{1.25a}{1.20b} = \frac{1.25}{1.20} \cdot \frac{a}{b} = \frac{1.25}{1.20} \cdot \frac{a}{b}. \]

Next, we calculate the factor \( \frac{1.25}{1.20} \): \[ \frac{1.25}{1.20} = \frac{125}{120} = \frac{25}{24}. \]

This means the new fraction in terms of the original fraction is: \[ \frac{25}{24} \cdot \frac{a}{b}. \]

To find the percent change in the fraction, we first calculate the change in value compared to the original fraction: \[ \text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100%. \]

Substituting the values we have: \[ \text{New Value} = \frac{25}{24} \cdot \frac{a}{b}, \quad \text{Original Value} = \frac{a}{b}. \]

Thus, \[ \text{Percent Change} = \frac{\frac{25}{24} \cdot \frac{a}{b} - \frac{a}{b}}{\frac{a}{b}} \times 100% = \left( \frac{25}{24} - 1 \right) \times 100%. \]

Calculating \( \frac{25}{24} - 1 \): \[ \frac{25}{24} - 1 = \frac{25}{24} - \frac{24}{24} = \frac{1}{24}. \]

Now substituting back into the percent change formula: \[ \text{Percent Change} = \left( \frac{1}{24} \right) \times 100% = \frac{100}{24}% \approx 4.1667%. \]

Thus, the fraction will change by approximately: \[ \boxed{4.17%}. \]