Question 901036
{{{2^2=4}}}
{{{r^2=5}}}
{{{3^2=9}}}


A simple way to find r is try to take averages and pick the approximation that gives result of 5 as close as you want.  Start with average of 2 and 3 and test this.


{{{(2+3)/2=5/2}}}.
{{{(5/2)^2=25/4=6&1/4}}}--- too high.


Find average of 2 and 5/2.
{{{(2+5/2)/2=(4/2+5/2)/2=9/4}}}.
{{{(9/4)^2=36/16=2&1/4}}}--- too low.


Try between 2.25 and 5/2.
(9/4+5/2)/2=(9/4+10/4)/2=19/(4*2)
{{{19/8}}}
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{{{(19/8)^2=5.640625}}}----too high

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Try 19/10.
{{{(19/10)^2=3.61}}}----too low


Next you can try something between 19/10 and 19/8, 
maybe try 19/9.
{{{(19/9)^2=4.4568}}}----too low.


You would want between 19/9 and 19/8;
Find their average, and test.
Average of 38/18 and 38/16....
continue this process if you want...


Another way is to use table of logarithms; which could be easy but "simple" depends on how comfortable you are with doing this.



Actually, 2.24 is fairly close to {{{sqrt(5).}}}
{{{2.24^2=5.0176}}}
and
{{{2.235^2=4.9952}}}