Question

A thief robs a house at 12 midnight and, as soon as he leaves the house owner realizes the robbery in the house. After 10 minutes, he rings the alert alarm, and the security guards of the house start running after the thief to catch him. If the speed of the thief is 30 km per hr and that of the security guards is 20 km per hr, what time will the guards take to catch the thief?
(a) 00.30 hrs
(b) 00.40 hrs
(c) 00.50 hrs
(d) Never catch the thief

Answer 1

Hint: In this question, we first need to find the distance that can be travelled by the thief in the interval of 10 minutes using the formula \[v=\dfrac{d}{t}\]. Then need to calculate the time taken by the securing guards to reach that distance. Now, as the thief travels some more distance in the interval the security guards take to reach the distance he travelled before we get the result according to that.

Complete step by step answer:
Now, given in the question that the speed of thief is 30 kmph and speed of security guards is 20 kmph
Let us assume the speed of the thief as v and speed of security guards as V
Given that
\[v=30,V=20\]
As we already know that the relation between speed, distance and time is given by the formula
\[v=\dfrac{d}{t}\]
Now, let us find the distance travelled by the thief in 10 minutes
Let us first convert minutes into hours as the speed is given in terms of hours
\[1hr=60\min \]
Now, according to the above conversion we get,
\[\Rightarrow 10min=\dfrac{10}{60}hr\]
Let us now substitute the respective values in the above formula
\[\Rightarrow 30=\dfrac{d}{\dfrac{10}{60}}\]
Now, on cross multiplication we get,
\[\Rightarrow d=30\times \dfrac{10}{60}\]
Now, on further simplification we get,
\[\Rightarrow d=5km\]
Let us now calculate the time taken by the security guards to travel this distance
\[\Rightarrow V=\dfrac{d}{T}\]
Now, on substituting the respective values we get,
\[\Rightarrow 20=\dfrac{5}{T}\]
Now, this can be further written as
\[\Rightarrow T=\dfrac{1}{4}hr\]
Now, on further simplification
\[\Rightarrow T=\dfrac{1}{4}\times 60\min \]
\[\therefore T=15\min \]
So, the security guards take more time than the time taken by the thief to travel a particular distance as their speed is less than that of the thief and they are starting late.
Thus, the guards cannot catch the thief
Hence, the correct option is (d).

Note:
Instead of calculating the time taken by the guards to travel a certain distance and then comparing it with that of the thief we can also solve this problem by considering the relative velocity and then conclude that as the guards and starting late they need to have higher velocity than that of the thief to catch him.
It is important to note that the time that we calculated which is taken by the guards is the time that they take the distance that they can travel which the thief travelled before they started chasing. Because this time when they reach there the thief travels further more which the guards won't travel.