Question : A car, during its entire journey of 5 hours, travels the first 45 minutes at a certain speed, the next 75 minutes at a speed of 85 km/hr, and the last 3 hours at a speed of 70 km/hr. During its entire journey, the average speed of the car is found to be 73 km/hr. What is the speed (in km/hr) of the car during the first 45 minutes?
Option 1: 65
Option 2: 68
Option 3: 62
Option 4: 72
Correct Answer: 65
Solution :
Let the speed in the first 45 minutes be $x$ km/hr and the total distance be $y$.
First 45 minutes = $\frac{45}{60}$ hours
= 0.75 hours
Next 75 minutes = $\frac{75}{60}$ hours
= 1.25 hours
Last 3 hours = 3 hours
Now,
Distance covered for first 45 minutes = $x\times 0.75$, where $x$ is the speed = $0.75x$
Distance covered for the next 75 minutes: Distance = $85 \times1.25$ = 106.25 km
And, for the last 3 hours: Distance = $70\times3$ = 210 km
We know,
Total distance = Distance in first 45 minutes + distance in next 75 minutes + distance in last 3 hours
= $0.75x+ 106.25 + 210$
And, Total time = 0.75 hours + 1.25 hours + 3 hours = 5 hours
Now,
Average speed = $\frac{\text{Total distance}}{\text{Total time}}$
⇒ $73 = \frac{0.75x+ 106.25 + 210}{5}$ hours
⇒ $365 = 0.75x + 106.25 + 210$
⇒ $365 = 0.75x + 316.25$
⇒ $48.75 = 0.75x$
⇒ $x = 65$ km/hr
Hence, the correct answer is 65.
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