Jimmy got a 38.5% (± 0.05%) on his math test. How many questions did the test have at a minimum?

Question

The answer is not "200 questions", though it would be if he got a score of exactly 38.5%. The fact that anything that rounds to the nearest decimal is allowed complicates things.

I know the answer, but can it be solved without brute force?

Answer 1

Continued fractions. $$0.385 = \frac1{2+\frac1{1+\frac1{1+\frac1{2+\frac1{15}}}}} $$ and $$ \frac1{2+\frac1{1+\frac1{1+\frac1{2}}}}=\frac5{13}\approx 0.3846$$ so it could have been $13$ questions.

It could not have bneen less because any fraction$\frac ab$ between $\frac38=0.375$ and $\frac25=0.4$ has $b\ge 13$: We have $\frac ab>\frac38$, which implies that the numerator of $\frac ab-\frac38=\frac{8a-3b}{8b}$ is $\ge1$ and on the other hand the numerator of $\frac25-\frac ab=\frac{2b-5a}{5b}$ is also $\ge1$, so that $$b= 5\cdot(8a-3b)+8\cdot(2b-5a)b\ge 5+8=13$$