Which is bigger $\frac{1}{{\sqrt{1001}}+ {\sqrt{1000}}}$ or $\frac {1}{10}$.

Question

What is the best way to find the inequality? I know the answer but my solution is messy.

$\sqrt{1001} + \sqrt{1000} > 10 + 10 > 10$. – William Stagner Apr 03 '15 at 22:38 — William Stagner, Apr 03 '15 at 22:38
Haha, yes. Thank you – William Stagner Apr 03 '15 at 22:39 — William Stagner, Apr 03 '15 at 22:39

score 2 · Accepted Answer · answered Apr 03 '15 at 22:52

2

Using the hint in the comments. Since \begin{align} \sqrt{1001} + \sqrt{1000} &>\sqrt{1000}+\sqrt{1000}\\ &=10\sqrt{10}+10\sqrt{10}\\ &=20\sqrt{10}\\ &>20 \end{align} We have that $$\frac{1}{\sqrt{1001}+\sqrt{1000}}<\frac{1}{20}<\frac{1}{10}.$$

answered Apr 03 '15 at 22:52

Eff

12,989

score 2 · Answer 2 · answered Apr 04 '15 at 14:03

2

Here's a very straightforward way of seeing the relation without doing any fancy manipulation: $$ \frac{1}{\sqrt{1001}+\sqrt{1000}}<\frac{1}{\sqrt{100}+\sqrt{100}}=\frac{1}{20}<\frac{1}{10}. $$

answered Apr 04 '15 at 14:03

Daniel W. Farlow

22,531

in fact, this inequality is so strong that I wonder if the original question was not in fact $\sqrt{101}+\sqrt{100}$.... 0.016 vs. 0.1 – MichaelChirico Apr 04 '15 at 20:21
@MichaelChirico I think you are probably right; otherwise, this inequality seems to be quite easy to establish. – Daniel W. Farlow Apr 04 '15 at 20:23
and, as your answer proves--quite easy to establish anyway (relatively speaking...) – MichaelChirico Apr 04 '15 at 20:24
1

@MichaelChirico Precisely. You may like the answer I provided for this question. Skip to the very bottom of my answer where I consider which one is bigger: $6^{1/4}$ or $4^{1/3}$. That's a pretty fun one! Much more interesting than the problem by OP. – Daniel W. Farlow Apr 04 '15 at 20:27

score 1 · Answer 3 · edited Apr 04 '15 at 20:29

1

With something like this it could help to start with approximation: $$ \sqrt{1000}\approx \sqrt{1001} \approx 30. $$

So your fraction is about $\frac{1}{60}$, which is plenty less than $\frac{1}{10}$.

Just saw that you already knew the direction, so here's how I'd formalize: $$ \frac{1}{\sqrt{1000}+\sqrt{1001}}< \frac{1}{\sqrt{900}+\sqrt{900}}=\frac{1}{60}<\frac{1}{10}. $$

edited Apr 04 '15 at 20:29

Daniel W. Farlow

22,531

answered Apr 04 '15 at 13:50

MichaelChirico

4,131

score -3 · Answer 4 · answered Apr 04 '15 at 01:13

-3

For the time being, leave the relation $@$ undefined. $\frac1{10}@\frac1{\sqrt{1001}+\sqrt{1000}}$
$\sqrt{1001}+\sqrt{1000}@10$
Compute $\sqrt{1001}$ and $\sqrt{1000}$.
Call their sum $x$.
$x@10$
Should $@$ be $\lt$, $=$ or $\gt$?

answered Apr 04 '15 at 01:13

Senex Ægypti Parvi

2,419

Which is bigger $\frac{1}{{\sqrt{1001}}+ {\sqrt{1000}}}$ or $\frac {1}{10}$.

4 Answers4