A man travels for 5 hours 15 minutes. If he covers the first half of the journey at 60km/h and rest at 45km/h. Find the total distance travelled by him. [SSC—CHSL (10 + 2) Exam, 2015]

### Concept & Formula When a journey is split into two equal halves (distances are the same) covered at different speeds $S_1$ and $S_2$, the relationship between total distance $D$ and total time $T$ can be evaluated by summing the time taken for each half. $$ \text{Total Time} = \frac{\text{Distance}_1}{\text{Speed}_1} + \frac{\text{Distance}_2}{\text{Speed}_2} $$ ### Step-by-Step Solution 1. **Convert Time:** The total time is 5 hours and 15 minutes. $$ 5 \text{ hours } 15 \text{ minutes} = 5 + \frac{15}{60} = 5 + \frac{1}{4} = \frac{21}{4} \text{ hours} $$ 2. **Define Distance:** Let the total distance be $2d$. Thus, the first half is $d$ and the second half is $d$. 3. **Set Up Equation:** The sum of times for both halves equals the total time. $$ \frac{d}{60} + \frac{d}{45} = \frac{21}{4} $$ 4. **Solve for $d$:** LCM of 60 and 45 is 180. $$ \frac{3d + 4d}{180} = \frac{21}{4} $$ $$ \frac{7d}{180} = \frac{21}{4} $$ $$ d = \frac{21 \times 180}{4 \times 7} $$ $$ d = 3 \times 45 = 135 \text{ km} $$ 5. **Calculate Total Distance:** $$ \text{Total Distance} = 2d = 2 \times 135 = 270 \text{ km} $$ ### Exam Strategy & Shortcut Use the average speed formula for equal distances: $\text{Average Speed} = \frac{2 \times S_1 \times S_2}{S_1 + S_2}$. Average Speed = $\frac{2 \times 60 \times 45}{60 + 45} = \frac{5400}{105} = \frac{360}{7} \text{ km/h}$. Total Distance = Average Speed $\times$ Total Time = $\frac{360}{7} \times \frac{21}{4} = 90 \times 3 = 270 \text{ km}$. ### Common Pitfall A common mistake is forgetting to multiply $d$ by 2 at the end if you assumed the total distance was $2d$. If you assume the total distance is $D$, the equation is $\frac{D/2}{60} + \frac{D/2}{45} = \frac{21}{4}$, which helps avoid this. ### Final Answer Therefore, the correct answer is **270 km**.