Difference between revisions of "1986 AJHSME Problems/Problem 7"
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Therefore, the smallest whole number between <math>\sqrt{8}</math> and <math>\sqrt{80}</math> is <math>3</math>. | Therefore, the smallest whole number between <math>\sqrt{8}</math> and <math>\sqrt{80}</math> is <math>3</math>. | ||
| − | Similarly, if we let <math>a=81</math>, <math>b=80</math>, and <math>c=64</math>, we get <cmath>\sqrt{81}>\sqrt{80}>\sqrt{64}</cmath> <cmath>9>\sqrt{80}> | + | Similarly, if we let <math>a=81</math>, <math>b=80</math>, and <math>c=64</math>, we get <cmath>\sqrt{81}>\sqrt{80}>\sqrt{64}</cmath> <cmath>9>\sqrt{80}>8</cmath> |
So <math>8</math> is the largest whole number between <math>\sqrt{8}</math> and <math>\sqrt{80}</math>. | So <math>8</math> is the largest whole number between <math>\sqrt{8}</math> and <math>\sqrt{80}</math>. | ||
Revision as of 20:36, 15 January 2009
Problem
How many whole numbers are between 
and 
?
Solution
No... of course you're not supposed to know what the square root of 8 is, or the square root of 80. There aren't any formulas, either. Approximation seems like the best strategy.
Clearly it must be true that for any positive integers 
, 
, and 
with 
, ![\[\sqrt{a}>\sqrt{b}>\sqrt{c}\]](http://latex.artofproblemsolving.com/8/f/2/8f205e911d02fc86594d46459f7fe1dcbc45d754.png)
If we let 
, 
, and 
, then we get ![\[\sqrt{9}>\sqrt{8}>\sqrt{4}\]](http://latex.artofproblemsolving.com/a/1/c/a1cbe429d4b72af1beee59a0cda479ddb1a5d7d0.png)
![\[3>\sqrt{8}>2\]](http://latex.artofproblemsolving.com/9/7/5/97547a3770496c9edf9d184444973b4f3771fd47.png)
Therefore, the smallest whole number between and is 
.
Similarly, if we let 
, 
, and 
, we get ![\[\sqrt{81}>\sqrt{80}>\sqrt{64}\]](http://latex.artofproblemsolving.com/f/5/3/f53152ce45564c5579eabf67dc635bd81d4b185d.png)
![\[9>\sqrt{80}>8\]](http://latex.artofproblemsolving.com/d/3/6/d36294b430ccc2caa40b5283dd32b7b6bcac8881.png)
So 
is the largest whole number between and .
So we know that we just have to find the number of integers from 3 to 8 inclusive. If we subtract 2 from every number in this set (which doesn't change the number of integers in the set at all), we find that now all we need to do is find the number of integers there are from 1 to 6, which is obviously 6.
