Juliet has attempted 213 problems on Brilliant and solved 210 of them correctly. Her friend Romeo has just joined Brilliant, and has 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.

Question

hero · Answer

I remember this question. Let's see if I can remember how to set it up.

hero · Answer

Hang on...let me check something. BRB

hero · Answer

Okay, I have it.

hero · Answer

Juliet's ratio: $$\frac{210}{213}$$

Romeo's ratio: $$\frac{2}{4}$$

Basically, the only way Romeo's ratio beats Juliet is if Juliet stops getting answers right

In other words:

$$\frac{2 + x}{4 + x} > \frac{210}{213 + x}$$

hero · Answer

Solve that for x

hero · Answer

Oh and $x$ equals the minimum number of problems they will attempt

hero · Answer

Probably should be $\ge$ sign

hero · Answer

This is the correct setup. All you need to do is solve for $x$. How is that going so far @marb0i

hero · Answer

Your solution will obviously involve a positive number. Discard any negative results.

anonymous · Answer

Is the answer a whole number? Beacause mine contains a decimal.

hero · Answer

Yes, the correct answer is a whole number. Mind sharing your steps? I can point you in the right direction.

hero · Answer

And then, what happened after that? You should post all your steps.

anonymous · Answer

$$\frac{ 2+x }{ 4+x } \ge \frac{ 210 }{ 213 + x }$$
$$(2 + x)(213 + x) = (4+x)(210)$$
$$426 + 2x + 213x +2x = 840 + 210x$$
$$426 - 840 = 210x - 217x$$
x =138

Am i doing this right?
sorry my net is so crappy

hero · Answer

No that is not correct. You have multiplied wrong

hero · Answer

There should be an $x^2$ in there somewhere.

hero · Answer

$426 + 2x + 213x +x^2 \ge 840 + 210x$

hero · Answer

That should be the correct inequality after cross multiplication