A boat, while going downstream in a river covered a distance of 50 miles at an average speed of 60 miles per hour. While returning, because of the water resistance, it took 1 hour 15 minutes to cover the same distance. What was the average speed during the whole journey?

### Concept & Formula Average speed for a round trip is the total distance divided by the total time. $$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$ ### Step-by-Step Solution - **Downstream Journey:** Distance = 50 miles. Speed = 60 mph. Time taken = $\frac{50}{60} = \frac{5}{6} \text{ hours}$. - **Upstream Journey:** Distance = 50 miles (since it covers the same distance). Time taken = 1 hour 15 minutes = $1\frac{15}{60} \text{ hours} = 1\frac{1}{4} = \frac{5}{4} \text{ hours}$. - **Total Journey:** Total Distance = $50 + 50 = 100 \text{ miles}$. Total Time = $\frac{5}{6} + \frac{5}{4} = \frac{10}{12} + \frac{15}{12} = \frac{25}{12} \text{ hours}$. - **Average Speed:** $$ \text{Avg Speed} = \frac{100}{\frac{25}{12}} = 100 \times \frac{12}{25} = 4 \times 12 = 48 \text{ mph} $$ ### Exam Strategy & Shortcut Alternatively, find the upstream speed: $50 / (\frac{5}{4}) = 40 \text{ mph}$. When distances are equal, the average speed formula is $\frac{2ab}{a+b}$ where $a$ and $b$ are the two speeds. $\frac{2 \times 60 \times 40}{60 + 40} = \frac{4800}{100} = 48 \text{ mph}$. ### Common Pitfall Calculating the arithmetic mean of the two speeds ($60 \text{ mph}$ and $40 \text{ mph}$) to get $50 \text{ mph}$. Average speed over equal distances requires the harmonic mean, not the arithmetic mean. ### Final Answer Therefore, the correct answer is **48 mph**.