A Brilliant member has solved 500 problems and got 350 of them right.

If he now starts getting every problem right, after how many problems will he have got 75% of all his problems right?

25
100
117
150

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When I solved this question, it said 75% of people got this right.

Abha Vishwakarma
- 2 years, 8 months ago

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So now one could ask how many of the next people should get it right to make the percentage exactly 75% again. Of course, if the next person gets it wrong, it will still be over 75%, but after two wrong people it might go under 75%. Interesting observation

Henry U
- 2 years, 8 months ago

It’s 73% now

Zoe Codrington
- 2 years, 7 months ago

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Unfortunately, it has been like this for quite some time. I don't think it will eventually stabilize at 75%, which would be funny.

Henry U
- 2 years, 7 months ago

I tried this with 3/4, not 0.75 and got x=-50. I just did trial and error after that

Zoe Codrington
- 2 years, 7 months ago

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You did something wrong then.

Simplifying (350 + x)/(500 + x) = 3/4 results in:

4(350 + x) = 3(500 + x)

Which further simplifies down to:

1400 + 4x = 1500 + 3x

From this, we get x = 100.

Faisal Mujawar
- 2 years, 5 months ago

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At my point of submission 75% of people got this right

Mohammad Farhat
- 2 years, 8 months ago

Lol, just saying - on a somewhat unrelated note - 75% of people also got this right,

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Actually, this is the third time somone is pointing that out. This brings me to the question, when the percentage of people is at 75% what is the exspected number of people who solve or don't solve it until the percentage reaches 75% again. Interesting!

Henry U
- 2 years, 8 months ago

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Keep in mind that if enough people solve this problem, it may stay at 75% for example 97/129 and 97/130 to the nearest whole percent are both 75%

(Incidentally, I got it right but it currently says 74%)

Jeremy Galvagni
- 2 years, 7 months ago

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Oh right, of course that makes it difficult

So at the moment, 301 out of 405 people solved it, which is – as you said – about 74%. But all 17 fractions $\frac{301+x}{405+x}, 3 \leq x \leq 19, x \in \mathbb{N}$ get rounded to 75%.

Henry U
- 2 years, 7 months ago

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@Henry U – I actually submitted a related problem a few months ago: https://brilliant.org/problems/possible-percent-problem/?ref_id=1491623

Jeremy Galvagni
- 2 years, 7 months ago

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@Jeremy Galvagni – Interesting problem!

Do you know a generalization to it? I'd imagine the solution to be almost the same for all values of the percentage because you're always looking for a fraction that lies or doesn't lie inside this intervall of lenght 1.

Henry U
- 2 years, 7 months ago

@Jeremy Galvagni – I tried to find a pattern in the values of $x$ (the smallest possible denominator), but it's difficult to make predictions about specific numbers. Now, I have also posted a problem on this topic.

Henry U
- 2 years, 7 months ago

Now it says 73%

Zoe Codrington
- 2 years, 7 months ago

(350+x)/(500+x)*100=75

Therefore x=100

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$\begin{aligned} \frac{350 + x}{500 + x} &= 0.75 \\ \\ 350 + x &= 375 + 0.75x \\ \\ 0.25x &= 25 \\ \\ &\boxed{x = 100} \end{aligned}$

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The percentage of right problems after solving $x$ more is given by

$\frac{350+x} {500+x}$ .

To get a percentage of 75% we set this equal to 0.75 and solve the resulting equation.

$\frac{350+x} {500+x} = 0. 75 \Leftrightarrow 350+x = 0.75 \cdot 500 + 0.75x \Leftrightarrow 0.25x = 25 \Leftrightarrow x = 100$