An algebra problem by A Former Brilliant Member

Algebra Level 3

The percentage chance of negativity exceeds that of positivity by a number which is the ratio of the two (former to later) .

What is the percentage chance of positivity that is close to $50\%$ ?

আরে! বাংলা কোথায় গেল?!!

The answer is 49.489688.

1 solution

Vinayak Srivastava
Sep 1, 2020

Let $\text{Chance(Positivity) }=x$ and so the $\text{Chance(Negativity) }=100-x$ (assuming that the number isn't 0).

We have the equation:

$100-x=x+\dfrac{100-x}{x}$

Multiplying, and simplifying, we get: $2x^2-101x+100 = 0$ Solving, $x = \dfrac{101 \pm \sqrt{101^2 - 4\cdot 2\cdot 100}}{4}$

Putting in our equation, we see that $x = \dfrac{101 + \sqrt{101^2 - 4\cdot 2\cdot 100}}{4}$ satisfies the condition, so it is the answer. $x = \dfrac{101 + \sqrt{101^2 - 4\cdot 2\cdot 100}}{4} \approx 49.48$

Your equation has minor mistake. It should be $2x^2 - 101x + 100 = 0$ :)

Aryan Sanghi - 9 months, 2 weeks ago

Thank you for pointing it out! I was also wondering why I had cancelled that $x$ !

Vinayak Srivastava - 9 months, 2 weeks ago

You did a mistake in solving the quadratic equation. It should be $4$ in the denominator. Check your final answer also. It is $\approx 49.489688$ .

A Former Brilliant Member - 9 months, 2 weeks ago

Oh sorry Sir. I had many typos and mistakes in the solution, is it ok now?

Vinayak Srivastava - 9 months, 2 weeks ago

Yes. Keep your cool while writing a solution. Have you appeared in RMO?

A Former Brilliant Member - 9 months, 2 weeks ago

@A Former Brilliant Member – No Sir. I came to know about these Olympiads only a few months ago, I will try to appear for PRMO and NSEJS this year if I can.

Vinayak Srivastava - 9 months, 2 weeks ago

An algebra problem by A Former Brilliant Member

The answer is 49.489688.

1 solution

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