A policeman chases a thief on a straight road. The policeman catches the thief in 15 minutes if he drives with a speed of 60 km/h and catches the thief in 0.5 hours if he drives with a speed 45 km/h. What should be his driving speed (in km/h) to catch the thief in just 10 minutes?

Let the constant speed of the thief be vkm/h and the initial separation distance between the policeman and the thief be dkm.
Case 1:
Policeman's speed = 60km/h
Time taken to catch the thief = 15minutes=
60
15
​
=0.25hours
Relative speed = (60−v)km/h
Distance equation:

d=(60−v)×0.25⟹4d=60−v⟹v=60−4d— (Equation 1)
Case 2:
Policeman's speed = 45km/h
Time taken to catch the thief = 0.5hours
Relative speed = (45−v)km/h
Distance equation:

d=(45−v)×0.5⟹2d=45−v⟹v=45−2d— (Equation 2)
Equating the two expressions for v:

60−4d=45−2d
60−45=42d−2d
15=2d⟹d=7.5km
Substitute d=7.5 back into Equation 2:

v=45−2(7.5)=45−15=30km/h
Case 3:
We need the policeman to catch the thief in 10minutes=
60
10
​
=
1
1
​
=
6
1
​
hours.
Let the required speed of the policeman be ukm/h.
Relative speed = (u−30)km/h

Distance equation:

d=(u−30)×
6
1
​

7.5=
6
u−30
​

7.5×6=u−30
45=u−30
u=45+30=75km/h
The required speed is 75 km/h.