= Solution
Let $E$ be the finish-time offset from the start of its displayed minute. The elapsed time is $25+E-S$, so $T=E-S$. To obtain the stated density, use the intended model that $E$ and $S$ are <independent random variables>, both with the <uniform distribution> on $[0,1)$.
Their joint <probability density function> is one on the unit square. Equivalently, the <convolution of probability densities> gives
$$
q(t)=\int_{\mathbb R}p(s)p(s+t)\,ds.
$$
The <integrand> is one exactly where $s\in[0,1]$ and $s+t\in[0,1]$. The overlap has length $1+t$ for $-1\leq t<0$, length $1-t$ for $0\leq t<1$, and zero otherwise. Therefore
$$
\boxed{q(t)=\begin{cases}1-|t|,&|t|<1,\\0,&|t|\geq1.\end{cases}}
$$
This is a symmetric <triangular distribution>. The independence and finish-time uniformity are additional modelling assumptions: the two truncated clock readings alone do not imply them. For example, if $E=S$ then the elapsed time is exactly twenty-five minutes and $T=0$, even though each endpoint offset is individually uniform.
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