Jim likes to keep track of his problem solving stats here on Brilliant. His target is a score (problems solved divided by problems attempted) of $80\%$ .

Initially, things don't go so well. He checks his score after a few weeks and finds it's well below his target.

After some more time doing courses, he checks again and is delighted to find his score is now above $80\%$ .

Was there necessarily a point in time when Jim's score was
*
exactly
*
$80\%$
?

[NB: this problem isn't original, but I've seen versions of it in too many places to give a single source]

Yes, definitely
No, not necessarily
Not enough information

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Let's say Jim never has a score of exactly $80\%$ . His score only increases when he gets a question correct, so at some point, his score must have jumped from below $80\%$ to above $80\%$ when he correctly answered a question.

Say the number of questions he had correctly answered before this was $c$ , and the number attempted $a$ . Then

$\frac{c}{a}<\frac45<\frac{c+1}{a+1}$

(just replacing $80\%$ with $\frac45$ .) From the left-hand inequality, we get $5c<4a$ . From the right-hand inequality, $4(a+1)<5(c+1)$ which becomes $4a<5c+1$ .

Putting these together, we have $5c<4a<5c+1$ . But all of these are integers; so the assumption that Jim's score is never exactly $80\%$ implies the existence of an integer between $5c$ and $5c+1$ . This contradiction means that, at some point, he must have had a score of exactly $80\%$ .

Bonus question: which other percentages have the same property?