By how much percent must a motorist increase his speed in order to reduce by 20%, the time taken to cover a certain distance?

### Concept & Logic This problem relies on the fundamental inverse relationship in kinematics: $$Distance = Speed \times Time$$ When the distance is kept constant, Speed and Time are inversely proportional. If time decreases by a certain fraction, speed must increase by the corresponding inverse fraction to maintain the exact same product. ### Step-by-Step Solution * **Initial Setup:** Let the original speed be $S$ and the original time be $T$. Distance = $S \times T$. * **Applying the Time Reduction:** The time is reduced by $20\%$. $20\%$ as a fraction is $\frac{1}{5}$. New Time = $T - (\frac{1}{5} \times T) = \frac{4}{5} \times T = 0.8T$. * **Determining the Required Speed:** Let the new speed be $S_{new}$. Since the distance is constant: $S_{new} \times (0.8T) = S \times T$ $S_{new} = \frac{S \times T}{0.8T} = \frac{1}{0.8} \times S$ $S_{new} = 1.25 \times S$. * **Calculating the Percentage Increase:** The new speed is $1.25$ times the original speed, which is $125\%$. Increase = $125\% - 100\% = 25\%$. ### Exam Strategy & Shortcut Use the fractional change shortcut. If a value $y$ decreases by $\frac{1}{n}$, to keep the product $x \times y$ constant, $x$ must increase by $\frac{1}{n-1}$. Time decreases by $20\%$, which is $\frac{1}{5}$. Here, $n = 5$. Speed must increase by $\frac{1}{5-1} = \frac{1}{4}$. $\frac{1}{4}$ expressed as a percentage is $25\%$. This takes about 3 seconds mentally. ### Common Pitfall The most universal mistake in inverse proportion problems is assuming that a $20\%$ decrease in one variable requires exactly a $20\%$ increase in the other variable. This is mathematically incorrect because the percentage increases are based on different starting bases. ### Final Answer **Therefore, the correct answer is 25.**