Question

Bonus Problem. By what percent will the fraction change if its numerator is increased by 25% and its denominator is increased by
20%?

Answer 1

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:
\[
\boxed{4.17\%}.
\]