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 % 50\% ?

আরে! বাংলা কোথায় গেল?!!


The answer is 49.489688.

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1 solution

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

We have the equation:

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

Multiplying, and simplifying, we get: 2 x 2 101 x + 100 = 0 2x^2-101x+100 = 0 Solving, x = 101 ± 10 1 2 4 2 100 4 x = \dfrac{101 \pm \sqrt{101^2 - 4\cdot 2\cdot 100}}{4}

Putting in our equation, we see that x = 101 + 10 1 2 4 2 100 4 x = \dfrac{101 + \sqrt{101^2 - 4\cdot 2\cdot 100}}{4} satisfies the condition, so it is the answer. x = 101 + 10 1 2 4 2 100 4 49.48 x = \dfrac{101 + \sqrt{101^2 - 4\cdot 2\cdot 100}}{4} \approx 49.48

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

Aryan Sanghi - 9 months, 2 weeks ago

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

Vinayak Srivastava - 9 months, 2 weeks ago

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

A Former Brilliant Member - 9 months, 2 weeks ago

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

Vinayak Srivastava - 9 months, 2 weeks ago

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

A Former Brilliant Member - 9 months, 2 weeks ago

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@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

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