Romeo and Juliet on Brilliant

Algebra Level 4

Juliet has attempted 213 problems on Brilliant and solved 210 of them correctly. Her friend Romeo has just joined Brilliant, and attempted 4 problems and solved 2 correctly. From now on, Juliet and Romeo will attempt all the same new problems. Find the minimum number of problems they must attempt such that it is possible that Romeo's ratio of correct solutions to attempted problems will be strictly greater than Juliet's.

The answer is 19.

2 solutions

Calvin Lin Staff
May 13, 2014

Currently, Romeo's ratio of correct solutions to attempted problems is $\frac{2}{4}$ , and Juliet's ratio is $\frac{210}{213}$ , so Juliet is far ahead. The minimum number of future problems they must both attempt so that Romeo's ratio will be greater than Juliet's occurs when Romeo gets all his solutions correct and Juliet gets none of her solutions correct. If they both attempt $n$ problems and Romeo gets them all correct and Juliet gets none of them correct, then Romeo's ratio becomes $\frac{2+n}{4+n}$ and Juliet's ratio becomes $\frac{210}{213 + n}$ . Therefore, we would like to find the smallest positive integer $n$ such that $\frac{2+n}{4+n} > \frac{210}{213+n}$ .

Multiplying both sides of this inequality by $(4+n)(213+n)$ gives
$(2+n)(213+n) > (210)(4+n), \mbox{ or } 426+215n+n^2 > 840 + 210n.$
This simplifies to $n^2+5n-414 > 0$ , and we can factor to obtain $(n-18)(n+23)>0$ . Therefore, for $0 \le n \le 18$ , Romeo's ratio is less than or equal to Juliet's ratio, but after attempting $19$ additional problems, it is possible for Romeo's ratio to be higher than Juliet's.

can you explain why you don't add n at 210 ????

Alfredo Sii Loebisx - 5 years, 11 months ago

You don't add an $n$ to $210$ because $210$ is the number of questions Juliet has gotten right. The fastest way for Romeo and Juliet to have the same ratio of questions correct to questions attempted is if Juliet gets all future questions wrong (meaning she has solved exactly $210$ questions out of $213+n$ ) and if Romeo gets all future questions right (meaning he will have solved $2+n$ questions out of $4+n$ ).

Trevor B. - 5 years, 11 months ago

What do you mean? At which particular step are you having trouble with?

Calvin Lin Staff - 5 years, 11 months ago

The question wasn't clear.I thought that they both get the same number of correct/wrong answers (Unless specified, it would be best to assume both their brains are equally matched). Similar to what Alfredo is asking, in which case the answer would be 414.

Vishnu Bhagyanath - 5 years, 11 months ago

@Vishnu Bhagyanath – It states "such that it is possible that", which means you have control over which problems a person gets right.

IE you are finding the minimum number of problems, such that it is possible in some scenario (given the history of problems solved), for Romeo's ratio to be strictly greater than Juliet's ratio

Calvin Lin Staff - 5 years, 11 months ago

@Calvin Lin – You have considered the best possible scenario, that is bit weird.

Som Ghosh - 5 years, 11 months ago

@Som Ghosh – It says "Find the minimum number of problems they must attempt such that it is possible ...". That is why we chose the best possible scenario.

Calvin Lin Staff - 5 years, 11 months ago

@Som Ghosh – If a question said "find the minimum number of problems you must attempt such that it is possible to get 3 correct" surely the answer is 3: attempting 0, 1 or 2 it would not be possible to get three correct but in the "best possible scenario", that you get them all correct, you have to attempt 3. Now you could go about arguing that it's "a bit weird" to consider such a scenario, let's assume you usually only get 50% correct on Brilliant, so then you must attempt 6. But that doesn't change the fact that it is POSSIBLE to get three correct answers with three problems, even if it is unlikely.

Joshua Prettyman - 5 years, 10 months ago

@Vishnu Bhagyanath – Answer in that case is 415. And I agree with you . I also thought the same.

Kushagra Sahni - 5 years, 10 months ago

For to decrease juliet's ratio we assume that she does not answer the questions correctly

Hamza Masood - 5 years, 11 months ago

Because you eant the smallest number, so you assume she will get none of them correctly and he gets all right... So you only add 'n' to 213

Gustavo Jambersi - 5 years, 7 months ago

so for Romio: total of correct questions= 210 totel of attempted questions = 213+ X

for Juliet: total no. of correct questions= 2+X totel of attempted questions = 4+ X given: 2+X / 4+x > 210/(213+ X) after solving..

X^2+ 5X - 414 > 0

(X - 18) (X +23) > 0

as x must be positive so X > 18

Mohamed IBrahim - 5 years ago

Great! When you answer the problem correctly, you can submit your solution directly :)

Calvin Lin Staff - 5 years ago

There's some kind of logical bias here, which makes it really hard to solve for most people. Can you name it?

Akshay Krishna - 2 years, 5 months ago

Can you be more explicit about what you think the logical bias is?

Are you referring to
1. Setting up the inequality
2. Figuring out what worst-case scenarios are

Calvin Lin Staff - 2 years, 5 months ago

Figuring out what worst-case scenarios are--there can be some conditional probability attested to the solution: assuming Juliet has a brain of her own and she has got 98.59% of brilliants problems correct earlier; you can't know for sure if she'll get those 19 problems--all wrong. (If it's not a bias, problems of this kind are more likely to cause certain hate for problem-solving as it stands up against the real world--that's my personal opinion)

Akshay Krishna - 2 years, 5 months ago

As usual your solution is brilliant and easy to understand.

But my doubt is that in the first line you have written "to attempted". Is it not grammatically incorrect?

I think it should be "to attempt".

Am i right?

Priyanshu Mishra - 5 years, 5 months ago

No it is correct. It says the ...ratio of 'correct solutions' to 'attempted questions'...

Ansh Bhatt - 5 years, 4 months ago

Right. The grammar rule that Priyanshu is thinking of is "If the sentence has the format of "To Verb", then the verb should be in present tense".

In this context is that the sentence is "The ratio of (Item A) to (Item B)", where Item A = correct solutions and Item B = attempted solutions.

Another similar example is saying that "I drove to (place X)", where X = bowling lessons (an activity). We do not say "I drove to bowl lessons".

Calvin Lin Staff - 5 years, 4 months ago

@Calvin Lin – Oh yes, it's correct. I didn't notice the ratio one.

Priyanshu Mishra - 5 years, 4 months ago

Brock Brown
Jun 30, 2015

Python 2.7:

j_solved = 210.0
j_attempts = 213.0
r_solved = 2.0
r_attempts = 4.0
juliet_ratio = j_solved / j_attempts
romeo_ratio = r_solved / r_attempts
problems = 0
while romeo_ratio <= juliet_ratio:
    # romeo solves a problem
    r_solved += 1
    r_attempts += 1
    # juliet fails to solve a problem
    j_attempts += 1
    juliet_ratio = j_solved / j_attempts
    romeo_ratio = r_solved / r_attempts
    problems += 1
print "Answer:", problems

Sledgehammer to crack a nut: programming a brute-force to substitute solving a quadratic equation

Daniel Heiß - 4 years, 10 months ago

Romeo and Juliet on Brilliant

The answer is 19.

2 solutions

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