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CAT 2020 Slot 1 — QA Question 1

TrainsEasy

Two persons are walking beside a railway track at respective speeds of 2 and 4 km per hour in the same direction. A train came from behind them and crossed them in 90 and 100 seconds, respectively. The time, in seconds, taken by the train to cross an electric post is nearest to

Answer & solution

  • A

    75

  • 82

  • C

    87

  • D

    78

Solution

Easy

When the train crosses a walking person, the relevant speed is the relative speed (train minus person). Each crossing gives the train's length LL in terms of its speed ss. Two crossings give two equations; eliminate LL to find ss, then the time to cross a post is L/sL/s.

1

Convert the walkers' speeds to m/s. Multiply km/h by 518\tfrac{5}{18}.

2 km/h=2×518=59 m/s4 km/h=4×518=109 m/s\begin{aligned} &2\text{ km/h}=2\times\tfrac{5}{18}=\tfrac{5}{9}\ \text{m/s}\\ &4\text{ km/h}=4\times\tfrac{5}{18}=\tfrac{10}{9}\ \text{m/s} \end{aligned}
2

Write each crossing as length = relative speed × time. Let the train's speed be ss m/s and length LL m.

L=(s−59)⋅90(first walker, 90 s)L=(s−109)⋅100(second walker, 100 s)\begin{aligned} &L=\left(s-\tfrac{5}{9}\right)\cdot 90 \quad\text{(first walker, 90 s)}\\ &L=\left(s-\tfrac{10}{9}\right)\cdot 100 \quad\text{(second walker, 100 s)} \end{aligned}
3

Equate and solve for ss. Set the two expressions for LL equal.

90s−50=100s−10009(from step 2)⇒ 10s=10009−50=5509⇒ s=559 m/s\begin{aligned} &90s-50=100s-\tfrac{1000}{9} \quad\text{(from step 2)}\\ &\Rightarrow\ 10s=\tfrac{1000}{9}-50=\tfrac{550}{9}\\ &\Rightarrow\ s=\tfrac{55}{9}\ \text{m/s} \end{aligned}
4

Find the length, then the post-crossing time. Substitute ss back into the first equation.

L=90s−50=90⋅559−50=550−50=500 mtime to cross a post=Ls=50055/9=450055=90011≈81.8 s\begin{aligned} &L=90s-50=90\cdot\tfrac{55}{9}-50=550-50=500\ \text{m}\\ &\text{time to cross a post}=\frac{L}{s}=\frac{500}{55/9}=\frac{4500}{55}=\frac{900}{11}\approx 81.8\ \text{s} \end{aligned}
90011≈82 seconds\dfrac{900}{11}\approx 82\ \text{seconds}
CAT 2020 Slot 1 QA Q1: Two persons are walking beside a railway track at respective speeds of 2 and 4 km per hour in the same directi — Solution | TheCATExam