# root estimation

Which of these is the best approximation for this square root?

Without using a calculator, work out which of the following is the best approximation for $$\sqrt{\frac{2001}{10000}}$$

$0.4,$ $0.42,$ $0.45,$ or $0.47$

$0.4,$ $0.42,$ $0.45,$ or $0.47$

*This problem is taken from the World Mathematics Championships***Answer**: 0.45

**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 $0.45$