In a fraction, if numerator is increased by 40% and denominator is increased by 80%, then what fraction of the original is the new fraction?

### Concept & Formula A fraction is composed of a Numerator ($N$) and a Denominator ($D$). When percentage changes are applied independently to the top and bottom, you simply multiply each component by its respective percentage multiplier. $$\text{New Fraction} = \frac{N \times (1 + \text{Numerator Increase \%})}{D \times (1 + \text{Denominator Increase \%})}$$ ### Step-by-Step Solution * **Initial Setup:** Let the original fraction be $\frac{N}{D}$. * **Applying the Changes:** The numerator is increased by $40\%$. New Numerator = $N + 0.40N = 1.40N$. The denominator is increased by $80\%$. New Denominator = $D + 0.80D = 1.80D$. * **Forming the New Fraction:** New Fraction = $\frac{1.40N}{1.80D}$. * **Simplifying the Ratio:** We need to find out what fraction of the *original* fraction this represents. New Fraction = $(\frac{1.40}{1.80}) \times (\frac{N}{D})$. Simplify the decimal multiplier: $\frac{1.40}{1.80} = \frac{14}{18}$. Divide the numerator and denominator by $2$: $\frac{14}{18} = \frac{7}{9}$. * **Conclusion:** The New Fraction is $\frac{7}{9}$ of the original fraction $\frac{N}{D}$. ### Exam Strategy & Shortcut You don't need algebra. Assume the original fraction is exactly $\frac{100}{100}$ (which is mathematically valid for finding ratios of change). New Numerator = $100 + 40 = 140$. New Denominator = $100 + 80 = 180$. New Fraction = $\frac{140}{180}$. Drop the zeros: $\frac{14}{18}$. Simplify by dividing by 2: $\frac{7}{9}$. The ratio of the new fraction to the old fraction is strictly $\frac{7}{9}$. ### Common Pitfall Students frequently choose "Data inadequate" because they assume they need to know the specific values of the original numerator and denominator to find the new fraction. However, the question asks "what *fraction of the original* is the new fraction" (a relative ratio), not the exact numerical value of the new fraction. ### Final Answer **Therefore, the correct answer is $\frac{7}{9}$.**