Time, Speed and Distance Aptitude basics, practice questions, answers and explanations 
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Important Formula and Equations

1. Speed, Time and Distance:
    Speed = Distance / time
    Time  = distance /speed
    Distance =speed*time
2. km/hr to m/sec conversion:
    x km/hr =[x*5/18] m/sec
3. m/sec to km/hr conversion:
    x m/sec =[x* 18/5] km/h
4. If the ratio of the speeds of A and B is a : b, then the ratio of the times taken by them to cover the same distance is 1/a:1/b or b : a
5. Suppose a man covers a certain distance at x km/hr and an equal distance at y km/hr. Then, the average speed during the whole journey is  [xy/(x+y)] km/hr

Key Note:

Caution average speed should not be calculated as average of different speeds, i.e., Ave. speed ≠(Sum of speed / No. of different Speed)
There are two different cases when average speed is required.

Case I

When time remains constant and speed varies :
If a man travels at the rate of x km/h for t hours and again at the rate of y km/h for another t hours, then for the whole journey, his average speed is given by
Average speed= Total distance/ Total time taken = (xt+yt)/(t+t) 
= (x+y)/2 kmph

Case II

When the distance covered remains same and the speeds vary :
When a man covers a certain distance with a speed of x km/h and another equal distance at the rate of y km/h. then for the whole journey, the average speed is given by Average speed =2xy/(x+y) km/h.

Velocity :The speed of a moving body is called as its velocity. If the direction of motion is also taken into consideration
Velocity = (Net displacement of the body)/(Time taken)

Relative speed:

a) Bodies moving in same direction
When two bodies move in the same direction, then the difference of their speeds is called the relative speed of one with respect to the other.
When two bodies move in the same direction, the distance between them increases (or decreases) at the rate of difference of their speeds.
b) Bodies moving in opposite direction
The distance between two bodies moving towards each other will get reduced at the rate of their relative speed (i.e., sum of their speeds).
Relative speed of one body with respect to other body is sum of their speeds.
Increase or decrease in distance between them is the product of their relative speed and time.

Key notes to solve problems

When a moving body covers a certain distance at x km/h and another same distance at the speed of y km/h, then average speed of moving body during its entire journey will be [2xy/(x+y)]km/h

A man covers a certain distance at x km/h by car and the same distance at y km/h by bicycle. If the time taken by him for the whole journey by t hours, then Total distance covered by him is equal to 2txy/(x+y) km.
A boy walks from his house at x km/h and reaches the school ' t1 ' minutes late. If he walks at y km/h he reaches ' t2 ' minutes earlier. Then, distance between the school and the house
=  ((xy)/(y-x))* (t1+t2)/60 km

If a man walks with (x/y) of his usual speed he takes t hours more to cover a certain distance, then the time to cover the same distance when he walks with his usual speed, (xt)/(y-x)  hours.

If two persons A and B start at the same time in opposite directions from the points and after passing each other they complete the journeys in 'x ' and ' y ' hrs. respectively, then A's speed: B's speed=speed=y:x

If the speed is (a/b) of the original speed, then the change in time taken to cover the same distance is given by Change in time = ((b/a)-1)*original time

Key notes to solve problems on Trains

The time taken by a train in passing a signal post or a telegraph pole or a man standing near a railway line = (Length of the train)/ (speed of the train) 
 
The time taken by a train passing a railway bridge or a platform or a tunnel or a train at rest= (x+y)/Speed  where, x = length of the train, y = length of the bridge or platform or standing train or tunnel

Time taken by faster train to pass the slower train in the same direction= (x+y)/(u-v); where, x = length of the first train ; y = length of the second train ; u = speed of the first train ; v = speed of the second train and u > v

Time taken by the trains in passing each other while moving in opposite direction =(x+y)/(u+v)

Time taken by the train to cross a man = x/(u-v) where, both are moving in the same direction and x= length of the train; u= speed of the train and v= speed of the man.

Time taken by the train to across a man running in the opposite direction= x/ (u+v)

If two trains start at the same time from two points A and B towards each other and after crossing, they take a and b hours in reaching B and A respectively. Then,  A's speed: B's speed= b: a

A train starts from a place at u km/h and another fast train starts from the same place after t hours at v km/h in the same direction. Find at what distance from the starting place both the trains will meet and also find the time of their meeting.
Distance= uvt/(v-u) km
Time=ut/(v-u) hours

The distance between two places A and B is x km. A train starts from A to B at u km/h. One another train after t hours starts from B to A at v km/h. At what distance from A will both the train meet and also find the time of their meeting
Time=(x-ut)/ (u+v) + t hours
Distance from A = u(((x-ut)/(u+v))+t) km

 
Two trains starts simultaneously from the stations A and B towards each other at the rates of u and v km/h respectively. When they meet it is found that the second train had traveled x km more than the first. Then the distance between the two stations
(i.e., between A and B)  is x(u+v)/ (v-u) km