A thief running at 7 km/h is chased by a policeman whose speed is 12 km/h. If the thief is 280 m ahead of the policeman, then the time required for the policeman to catch the thief will be:

Correct Answer: $3\frac{9}{25}\text{minutes}$

Step-by-step explanation:

Step 1: Identify the given values:

Speed of the thief = 7 km/h

Speed of the policeman = 12 km/h

Initial distance between the thief and the policeman = 280 m

Step 2: Calculate the relative speed:

Since both the policeman and the thief are moving in the same direction, the relative speed of the policeman with respect to the thief is the difference between their speeds:

$\text{Relative Speed}=12\text{km/h}-7\text{km/h}=5\text{km/h}$

Step 3: Convert relative speed to meters per minute:

Since 1 km = 1000 meters and 1 hour = 60 minutes:

$\text{Relative Speed in m/min}=5\times \frac{1000}{60}\text{m/min}=\frac{500}{6}\text{m/min}=\frac{250}{3}\text{m/min}$

Step 4: Calculate the time required to catch the thief:

Using the formula for time:

$\text{Time}=\frac{\text{Distance}}{\text{Relative Speed}}$

Substitute the distance (280 m) and relative speed ($\frac{250}{3}$ m/min):

$\text{Time}=\frac{280}{250/3}\text{minutes}$

$\text{Time}=280\times \frac{3}{250}\text{minutes}$

$\text{Time}=\frac{28\times 3}{25}\text{minutes}=\frac{84}{25}\text{minutes}$

Step 5: Convert the improper fraction to a mixed fraction:

Dividing 84 by 25 gives 3 with a remainder of 9:

$\frac{84}{25}=3\frac{9}{25}\text{minutes}$

Hence, the time required for the policeman to catch the thief is $3\frac{9}{25}\text{minutes}$.