Tweaking a Simple Division

Algebra Level 3

What positive real number, when divided by one less than itself, gives one more than itself?

Give your answer to 3 decimal places.

The answer is 1.618.

2 solutions

Andrew Ellinor
Jan 28, 2016

Let $x$ be this real number. The equation we're looking to solve is

$\frac{x}{x - 1} = x + 1$

Multiplying both sides by $x - 1$ and subtracting $x$ over to the right side gives $x^2 - x - 1 = 0.$ Using the quadratic formula , we find our answer to be

$x = \frac{1 \pm \sqrt{5}}{2}$

Because $x$ is to be positive, our answer is $\boxed{1.618}.$

Challenge taking student note: What positive real number, when divided by one more than itself, gives one less than itself?

Nihar Mahajan - 5 years, 4 months ago

It will be the same answer.

Rajdeep Bharati - 5 years, 4 months ago

You can just cross divide, you will obtain the same answer.

Ashish Menon - 5 years, 2 months ago

Sachin Sharma
Feb 1, 2016

Phi ( $Φ = 1.618033988749895…$ ), is simply an irrational number like pi $(π)$ , but one with many unusual mathematical properties. Unlike $π$ , which is a transcendental number, $Φ$ is the solution to a quadratic equation

Φ² = Φ + 1 or 1 / Φ = Φ – 1

It is the basis for the Golden Ratio, Section or Mean

Tweaking a Simple Division

The answer is 1.618.

2 solutions

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