1986 AJHSME Problems/Problem 7
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 ,
If we let , , and , then we get
Therefore, the smallest whole number between and is .
Similarly, if we let , , and , we get
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.
See Also
1986 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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All AJHSME/AMC 8 Problems and Solutions |
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