Relativity of time I have considered the special relativity and time dilation equations. I found that time dilation in clock moving with velocity (v) observed from rest is $T'=T(\sqrt{1-v^2/c^2}) $and time dilation in clock at rest observed from moving body is $T'=T/(\sqrt{1-v^2/c^2})$. Are these equation holds good in their respective frame of reference? If yes, why time does not increases for one and decreases for other with the same rate. 
Thank you in advance.
 A: Let me try to give an intuitive explanation.
The way each person reads the other's clock is to send out a radar pulse, which reflects back and contains the other person's clock reading.
Then, each one considers the other's clock to have had that reading half way between sending and receiving the radar pulses.
I think it helps to visualize a space-time diagram, like this:

The units of time are years, and the units of X are light-years, so the 45-degree line represents the speed of light.
There are two people, A and B, and just for simplicity, they start at 0,0 and travel in opposite directions at high speed (but less than the speed of light).
Each one sends out radar pulses (shown in light gray) to read the other's clock.
You can see that at time 1.2, A reads B's clock as 1. Similarly at time 1.2, B reads A's clock as 1.
(What each considers to be "the same time" is shown in blue.)
So each thinks the other's clock is running slow.
A: 
and time dilation in clock at rest observed from moving body is
  $T′=T/\sqrt{1−v^2/c^2}$.

It isn't quite clear to me what you're thinking about here when you write "a clock at rest as observed from a moving body".
Let's say that Alice observes that Bob is moving uniformly.  What we mean is that, in an inertial coordinate system in which Alice's position is constant with time, Bob's position is changing uniformly with time.
So, in Alice's coordinates, Bob is moving.
But motion is relative.
In an inertial coordinate system in which Bob's position is constant with time, Alice's position is changing uniformly with time and thus, in Bob's coordinates, Alice is moving.
Clearly, a clock at rest in Alice's coordinates is moving in Bob's coordinates.  To observe the rate at which Alice's clock runs, Bob must use two spatially separated clocks, both at rest and synchronized in Bob's coordinates.
It's easy to show, via the Lorentz transformations that relate Alice's and Bob's coordinates, that Bob finds Alice's clock runs slower than clocks at rest in Bob's coordinates.
However, exchanging the names Alice and Bob in the above two paragraphs does not change the result.
That is, Bob (Alice) finds that Alice's (Bob's) clock runs slowly compared to clocks at rest in Bob's (Alice's) coordinates.
In summary, I don't know what to make of "a clock at rest as observed from a moving body".
A clock at rest with respect to Alice (Bob) as observed by a moving Bob (Alice) is not at rest according to Bob (Alice).
Put another way:  "a clock at rest [at rest according to whom?] as observed from a moving body [moving according to whom?]".
A: Suppose we have a clock in a moving spaceship which is cruising at 99.995% of the speed of light (299,777,468.3771 metres/sec) on a journey of 200 light-years;
using the following equation:
$$
γ = {1\over \sqrt{1-(v/c)^2}}
$$
$γ = 100.$
This means that an observer with a clock at rest will find that the moving clock is moving slower by the $γ$ (gamma) factor which is 100.
So for example, if several successive generation of observers at rest on Earth will observe the spacecraft for the duration of its journey, they will find that the spacecraft's clock is moving really slowly to the point that though 200 years will have passed on earth yet the spacecraft's clock will record only 2 years.
However the spacecraft's occupants will find that the clocks on earth are moving very fast in comparison to their own on-board clocks by the same gamma factor.
Your equations should be as follows:


*

*time dilation in clock moving with velocity (v) observed from rest is


$$
T' = t \sqrt{1-(v/c)^2}
$$
and 


*

*time in clock at rest observed from moving body is


$$
t = {T'\over \sqrt{1-(v/c)^2}}
$$
Time is being dilated only for the moving observer, however the time in the rest frame is running normally.
A: The equation is useful for calculating the value of the discrepancy but it doesn't tell you the sign (which frame is the one the experienced more/less time). To find that out you'd need to know which inertial frame is the one that changed in reference to the other (accel/deceleration is not relative).
