Suppose that there were $n$ questions on the test.
Since Chris received a mark of $50 \%$ on the test, then he answered $\frac{1}{2} n$ of the questions correctly. We know that Chris answered $13$ of the first $20$ questions correctly and then $25 \%$ of the remaining questions.
Since the test has $n$ questions, then after the first $20$ questions, there are $n-20$ questions. Since Chris answered $25 \%$ of these $n-20$ questions correctly, then Chris answered $\frac{1}{4}(n-20)$ of these questions correctly.
The total number of questions that Chris answered correctly can be expressed as $\frac{1}{2} n$ and also as $13+\frac{1}{4}(n-20)$.
Therefore, $\frac{1}{2} n=13+\frac{1}{4}(n-20)$ and so $2 n=52+(n-20)$, which gives $n=32$.
(We can check that if $n=32$, then Chris answers $13$ of the first $20$ and $3$ of the remaining $12$ questions correctly, for a total of $16$ correct out of $32.)$
