Your question is far from trivial. You can't imagine how many people are confused as you are. I'll do my best to explain the reason behind that.
My answer to a related question, a few days ago, began:
First of all, I think you should learn the basic concepts for this
kind of problems. There are three:
Let's apply my prescription to your problem.
Objects. The problems refers to a beam of muons, but it is better to think of a single muon. This is the object.
Events. There are two: the muon's birth $B$ and its decay $D$.
Reference frames. Two frames are involved: the muon's rest frame $K$ (problem's text says "proper frame") and the laboratory frame $K'$.
Now for time measurements. No problem in frame $K$, where the muon is at rest. One clock $C$ is enough, and $\tau$ is the time elapsed from event $B$ to event $D$, as recorded by $C$. Note that we can safely assume that $C$ is placed just near the resting muon.
Measuring time in $K'$ is another story: since the muon is moving in this frame, one clock is not enough. This was Einstein's main intuition: unless you assume there is an absolute time, when you are interested in events happening in different places of your reference frame you must read times via two clocks, each located in the place where an event happens. (Of course, clocks must have been previously synchronized. I cannot dwell explaining how synchronization may be effected).
Actually clocks are not all, as you have to detect emission $B$ and decay $D$ of the muon. So in two places you will set two more complex instruments, $I_1$ and $I_2$, able to detect the relevant events and record their times. Then you have only to read these times and calculate the difference $\tau'$. You expect this difference equates $\gamma\,\tau$.
After that, you begin another story, which talks of aging people. But I urge you to apply my prescription again:
- Which is the object? It is the aging person $P$, no longer the muon.
- Which are the frames? (I exchanged the order of questions for reasons you will presently understand.) They are $K$ and $K'$, as before, but the person is resting in $K'$, not in $K$.
- Which are the events? Here you have to decide: Where do you want to put the person? Suppose you choose to put him near $I_1$. Then the first event can coincide with $B$ (muon's emission) but the second cannot coincide with $D$, which in $K'$ happens elsewhere. Take whatever event you like in $P$'s life, say $F$, and call $\theta'$ the time interval (measured by $I_1$) from $B$ to $F$.
Surely you may take measurements on $B$ and on $F$ in frame $K$, but in this frame these events do not happen in the same place, so two clocks are needed. If $\theta$ is the time interval as measured by these clocks, which relation do you expect between $\theta$ and $\theta\,'$?
I am sure you see that now the situation is exactly reversed wrt the muon case: events $B$ and $F$ (relating to $P$) happen in the same place of $K'$ but in different places of $K$. Therefore time dilation acts in reverse: $\theta=\gamma\,\theta'$. There's no contradiction, since the $\tau$'s referred to events $B$ and $D$, whereas the $\theta\,$'s refer to $B$ and $F$.