Approximating Years

Algebra Level 2

To which whole number is the value below closest?

$\Huge \frac{\color{#3D99F6}{2014} \times \color{#D61F06}{2015}^{2}}{\color{#20A900}{2016}^{2}}$

2015 2012 2014 2013

2 solutions

Eli Ross Staff
Sep 18, 2015

We can write this as $2014 \cdot \left(\frac{2015}{2016}\right)^2 = 2014 \cdot \left(1-\frac{1}{2016}\right)^2.$

Expanding out the squared term gives

$2014 \cdot \left(1 - 2\cdot \frac{1}{2016} + \frac{1}{2016^2} \right)= 2014 - 2\cdot \frac{2014}{2016} + 2014 \cdot \frac{1}{2016^2} \approx 2014 - 2 + 0 = 2012.$

Interesting problem! Can you explain that second step for me real quick though? I think I'm missing out on a rule I should know!

John Witter - 5 years, 8 months ago

2016^2 tends to infinity. So 1/(2016^2) tends to 0

Debayudh Rarhi - 5 years, 8 months ago

second step is the simple expansion of (1 - 1/x)^2

Poulastya Ray - 5 years, 8 months ago

2015/2016 = 2016/2016 - 1/2016 = 1 - 1/2016

Jess Doe - 5 years, 4 months ago

How: (2 2014)/2016 = 2 and 2014/(2016 2016) = 0 can you please explain it?

Hassan Shahzad - 5 years, 8 months ago

Close to 2 and close to 0. That's it.

Zakir Dakua - 5 years, 8 months ago

2x(2014/2016) close to 2*(1), so 2x(2014/2016) close to 2.
2014/(2016x2016) close to 2014/(2014x2016) close to 1/2016 close to 0 .

Prasit Sarapee - 5 years, 8 months ago

But how: In response to Hassan Shahzad: 2x(2014/2016) = 2 and 2014/(2014x2016) =0

Hassan Shahzad - 5 years, 8 months ago

@Hassan Shahzad – .

(2014/2016) close to 1, so 2x(2014/2016) close to 2x(1) = 2
2014/(2014x2016) = 1/2016 and 1/2016 close to 0,
so 2014/(2014x2016) =1/2016 close to 0.

Prasit Sarapee - 5 years, 8 months ago

@Hassan Shahzad – Approximate

Kobe Cheung - 5 years, 4 months ago

Renato Silva
Sep 19, 2015

Calculator!!!

Hahaha LOL

Athieccha Ufiie Nurmufidah - 5 years, 8 months ago

Approximating Years

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