Question : A thief steals an item and escapes, running at 20 km/hr. A policeman arrives at the spot of the crime after 6 minutes and immediately starts chasing the thief. 24 minutes after the policeman started to chase the thief, there is still a gap of 400 m between the two. At what distance from the spot of crime would the policeman catch up with the thief and what is the speed at which the policeman ran?
Option 1: 15 km; 25 km/hr
Option 2: 14.4 km; 24 km/hr
Option 3: 10 km; 25 km/hr
Option 4: 12 km; 24 km/hr
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Correct Answer: 12 km; 24 km/hr
Solution :
Speed of the thief = 20 km/hr
Let the speed of the policeman be $x$ km/hr.
Time after the policeman starts chasing the thief = 6 minutes
Distance between them 30 minutes after the crime = 400 m = 0.4 km
According to the question,
$20\times \frac{30}{60} - x\times \frac{24}{60} = 0.4$
$⇒100 - x\times 4 = 4$
$\therefore x = 24$ km/hr
Relative speed of policeman = 24 km/hr – 20 km/hr = 4 km/hr
So, the time after which the thief is caught (after 30 minutes of stealing) = $\frac{\text{Relative distance}}{\text{Relative speed}}$ = $\frac{0.4}{4}$ = 0.1 hr
Total distance from the crime scene where thief gets caught = Speed of thief × time for which thief ran = $20\times (\frac{30}{60}+0.1)=12$ km
Hence, the correct answer is 12 km; 24 km/hr.
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