0
$\begingroup$

Determine which one of the following two numbers is greater: $\frac{2.00013579}{1.00013579^2+2.00013579}$ or $\frac{2.0002468}{1.0002468^2 + 2.0002468}$.

How can I do this without a calculator? I tried comparing the numerator to the numerator, and the denominator to the denominator, but that did not work. I tried seeing whether the squares would change very little since they were really close to $1$, but I couldn't form any solid reasoning.

Thanks!

$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

Hint: let $2.00013579=x,2.0002468=y$,$x,y> 2$

Also notice that $$f(x)=\frac{x}{{(x-1)}^2+x}=\frac{1}{x+\frac{1}{x}-1}$$ is decreasing for $x\ge 1$

As $x<y$:$$\frac{x}{{(x-1)}^2+x}>\frac{y}{{(y-1)}^2+y}$$

$\endgroup$
6
  • $\begingroup$ Thank you! :) This helped a lot. $\endgroup$
    – Mike Smith
    Commented Nov 12, 2020 at 6:56
  • $\begingroup$ @MikeSmith you are welcome! $\endgroup$
    – Albus Dumbledore
    Commented Nov 12, 2020 at 6:58
  • $\begingroup$ you could say it's decreasing on $[1, \infty)$. $\endgroup$
    – orangeskid
    Commented Nov 12, 2020 at 7:21
  • $\begingroup$ @orangeskid ofcourse i was thinking of that but it does not matter here ,does it? $\endgroup$
    – Albus Dumbledore
    Commented Nov 12, 2020 at 7:22
  • 1
    $\begingroup$ I guess so... once you have the nice expression for $f(x)$ it's all clear $\endgroup$
    – orangeskid
    Commented Nov 12, 2020 at 7:27

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .