CAT Practice : Speed Time, Races

If you are travelling to Delhi and your car breaks down, you should thank your lucky stars, take the hint and go back home.

Speed Time

Q.5: Mr. X decides to travel from Delhi to Gurgaon at a uniform speed and decides to reach Gurgaon after T hr. After 30 km, there is some engine malfunction and the speed of the car becomes ${ {4 \over 5} ^{th}}$ of the original speed. So, he travels the rest of the distance at a constant speed ${ {4 \over 5} ^{th}}$ of the original speed and reaches Gurgaon 45 minutes late. Had the same thing happened after he travelled 48 km, he would have reached only 36 minutes late. What is the distance between Delhi and Gurgaon?
1. 90 km
2. 120 km
3. 20 km
4. 40 km

Choice B. 120 km

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Detailed Solution

Let the distance from Delhi to Gurgaon be ‘d’ km. The first 30 km he travels at his usual speed. However, the remaining ‘d-30’ km he travels at a reduced speed.

To travel ‘d’ km he usually takes T hr. Therefore, to travel ‘d - 30’ km he should ideally take ${(d - 30) x T \over d}$hr. However, this is only if he travels at his usual speed. It is given that he traveled only at ${ {4 \over 5} ^{th}}$ of his usual speed. Because of this he would have taken ${ {5 \over 4} ^{th}}$ of the time to travel the remaining distance, i.e., he takes ${ {1 \over 4} ^{th}}$ of the time extra. This is given to be 45 minutes (or ${ {3 \over 4} ^{th}}$hr)

${{1 \over 4} {x} {(d - 30) x T \over d} {=} {3 \over 4}}$      ........(1)

On the other hand, had the same thing happened after he travelled 48 km, he would have reached only 36 minutes or ${3 \over 4}$ hrs late. Hence,
${{1 \over 4} {x} {(d - 48) x T \over d} {=} {3 \over 5}}$       ........(2)

Dividing (1) by (2) and solving for d, we get d = 120 km.

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"Life is a race ... if you don't run fast ... you will be like a broken undaa" - Veerusahasra Buddhi