Ratio of speed of Bus and Train = 15 : 27.
Let speed of the bus is 15X and Speed of the Train is 27X.
Car Covered 720 km in 9 hours.
So, Speed of the Car = ^{720}/_{9} = 80 kmph.
Given, Speed of the bus is ^{3}/_{4} of Car, So speed of the Bus,
= ^{(80*3)}/_{4} = 60 kmph.
Thus,
15X = 60
X = 4
So, Speed of the train = 27X = 27*4 = 108 kmph.
Hence, Train will cover distance in 7 hours,
= 108 * 7 = 756 km.
P__150m___T______X m (Let)_______Q.
Let Policeman caught thief at a distance (X + 150)m. And Thief has traveled X m.
Speed of Policeman = 12 kmph = ^{(12*5)}/_{18} = ^{60}/_{18} m/sec.
Speed of thief = 10 km = ^{(10*5)}/_{18} = ^{50}/_{18} m/sec.
In this case time is constant means Policeman covered (X + 150)m in same time thief covered X m.
Thus,
Speed of the thief /Speed of Policeman = ^{X}/_{(150+X)} ^{50}/_{60} = ^{X}/_{(150+X)}
7500 + 50X = 60X
10X = 7500
X = 750 m.
So, Thief has traveled 750 m before the caught.
Let usual speed be X kmph, then new speed will be (X+400)kmph.
Time taken to cover 1600 km with speed X kmph,
= ^{1600}/_{X}
Time taken to cover 1600 km with Speed (X+400) kmph,
= ^{1600}/_{(X+400)}
Now,
Time difference = 40 minutes.
(^{1600}/_{X})-(^{1600}/_{(X+400)}) = ^{40}/_{60} hours.
X^{2} +400X - 960000 = 0
On solving,
X = -1200, 800.
Speed cannot be negative, So usual speed will be 800 km per hour.
S1=X4andS2=X5Required time to cross each other,={2X[(X4)+(X5)]}=409seconds$$\begin{array}{rl}& {S}_{1}=\frac{X}{4}\phantom{\rule{thinmathspace}{0ex}}\text{and}\\ & {S}_{2}=\frac{X}{5}\\ & \text{Required time to cross each other,}\\ & =\left\{\frac{2X}{\left[\left(\frac{X}{4}\right)+\left(\frac{X}{5}\right)\right]}\right\}\\ & =\frac{40}{9}\text{seconds}\end{array}$$
He proceeds at ^{4}/_{5} S where S is his usual speed means ^{1}/_{5} decrease in speed which will lead to ^{1}/_{4} increase in time. Now the main difference comes in those 12km (30-18) and the change in difference of time = (45-36) min = 9 min
Thus, ^{1}/_{4} * T = 9 min where T is the time required to cover the distance of (30 - 18) = 12 km T = 36 min = ^{36}/_{60} hours = 0.6 hours.