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.)