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Which of these five algebraic expressions is largest, given $x$ is between 0 and 1?

# Root Estimation

##### Age 14 to 16 ShortChallenge Level

Estimating the root
$10000=100^2$, so $\dfrac{2001}{10000} \approx \left(\dfrac{?}{100}\right)^2$
$?^2\approx 2001$
$40^2=1600$ too small
$50^2=2500$ too big
$45^2=2025$ close but too big
$44^2=1936$ close but not as close
So $\dfrac{2001}{10000}\approx\left(\dfrac{45}{100}\right)^2$
$\therefore \sqrt{\dfrac{2001}{10000}}\approx\dfrac{45}{100}=0.45$

Squaring the options
$\dfrac{2001}{10000}=0.2001$

$0.4^2=0.16$ too small
$0.42^2=0.1764$ too small
$0.45^2=0.2025$ too big
So $0.47^2$ will be too big too

$\sqrt{0.2001}$ is between $0.42$ and $0.45$. We should check numbers between $0.42$ and $0.45$ to find whether it is closer to $0.42$ or $0.45$

$0.2025$ closer than $0.1764$ to suspect $0.45$ closer so try $0.44$:
$0.44^2=0.1936$ too small
So $\sqrt{0.2001}$ is between $0.44$ and $0.45$
The option is it closest to is $0.45$

You can find more short problems, arranged by curriculum topic, in our short problems collection.