Problem 268

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 $2n = 52+(n-20)$, 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.)