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CAT 2020 Slot 1 — QA Question 4

Basics of TSD/ProportinalityEasy

A train travelled at one-thirds of its usual speed, and hence reached the destination 30 minutes after the scheduled time. On its return journey, the train initially travelled at its usual speed for 5 minutes but then stopped for 4 minutes for an emergency. The percentage by which the train must now increase its usual speed so as to reach the destination at the scheduled time, is nearest to

Answer & solution

  • A

    58

  • B

    61

  • 67

  • D

    50

Solution

Easy

Speed and time are inversely proportional over a fixed distance. The outward trip pins down the usual one-way time. On the return, subtract the 5 minutes already run and the 4-minute halt to find how little time is left for the rest, then convert that time-ratio into the required speed increase.

1

Find the usual travel time tt. At 13\tfrac{1}{3} speed the time triples, and the extra time is 30 minutes.

3t−t=30⇒ 2t=30⇒ t=15 minutes\begin{aligned} &3t-t=30\\ &\Rightarrow\ 2t=30\\ &\Rightarrow\ t=15\ \text{minutes} \end{aligned}
2

Account for the return-trip events. The first 5 minutes are run at usual speed, covering the part that usually takes 5 minutes; 44 minutes are lost to the halt. To still arrive on the 15-minute schedule, the remaining stretch (which usually needs 15−5=1015-5=10 minutes) must now be done in the leftover time.

time left for the rest=15−5−4=6 minutesusual time for that stretch=10 minutes\begin{aligned} &\text{time left for the rest}=15-5-4=6\ \text{minutes}\\ &\text{usual time for that stretch}=10\ \text{minutes} \end{aligned}
3

Convert the time ratio into a speed increase. Time shrinks to 610\tfrac{6}{10}, so speed scales by the reciprocal 106=53\tfrac{10}{6}=\tfrac{5}{3}.

new speed=53×(usual)⇒ % increase=(53−1)×100=23×100≈66.7%\begin{aligned} &\text{new speed}=\frac{5}{3}\times(\text{usual})\\ &\Rightarrow\ \%\text{ increase}=\left(\frac{5}{3}-1\right)\times 100=\frac{2}{3}\times 100\approx 66.7\% \end{aligned}
≈67%\approx 67\%
CAT 2020 Slot 1 QA Q4: A train travelled at one-thirds of its usual speed, and hence reached the destination 30 minutes after the sch — Solution | TheCATExam