# Time and Distance Shortcut Methods

The terms time and distance are related to the speed of a moving object.
Speed: Speed is defined as the distance covered by an object in unit time.

Some Important Facts

Distance travelled is proportional to the speed of the object if the time is kept constant.
Distance travelled is proportional to the time taken if speed of object is kept constant.
Speed is inversely proportional to the time taken if the distance covered is kept constant.
If the ratio of two speeds for same distance is a:b then the ratio of time taken to cover the distance is b:a
Relative Speed
If two objects are moving in same direction with speeds of x and y then their relative speed is (x - y)
If two objects are moving is opposite direction with speeds of x and y then their relative speed is (x + y)
Rule 1:

If some distance is travelled at x km/hr and the same distance is travelled at y km/hr then the average speed during the whole journey is given by

Example
John goes from his home to school at the speed of 2 km/hr and returns at the speed of 3 km/hr. What is his average speed during whole journey in m/sec?
Sol:
Let’s say x = 2 km/hr
And y = 3 km/hr, so

Now, average speed in m/sec

Rule 2:

If a person travels a certain distance at x km/hr and returns at y km/hr, if the time taken to the whole journey is T hours then the one way distance is given by

Example
Mr Samson goes to market at the speed of 10 km/hr and returns to his home at the speed of 15 km/hr. If he takes 3 hours in all, what is the distance between his home and market?
Sol:
Let’s say x = 10 km/hr
y = 10 km/hr, and
T = 3 hrs, then

So the distance between home and market is 18 km.
Rule 3:

If two persons A and B start their journey at the same time from two points P and Q towards each other and after crossing each other they take a and b hours in reaching Q and P respectively, then

Example
Two persons Ram and Lakhan start their journey from two different places towards each other’s place. After crossing  each other, they complete their journey in 1 and 4 hours respectively. Find speed of Lakhan if speed of ram is 20 km/hr.
Sol:
Let’s say A = Ram and B = Lakhan
a = 1 and b = 4, then

Lokhan's speed = 10km/hr

Rule 4:

If the same distance is covered at two different speeds S1 and S2 and the time taken to cover the distance are T1 and T2, then the distance is given by

Example
Two trucks travel the same distance at the speed of 50 kmph and 60 kmph. Find the distance when the distance when the time taken by both trucks has a difference of 1 hour.
Sol:
Let’s say S1 = 50 kmph,
S2 = 60 kmph
T1 – T2 =  1

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