Difference between revisions of "2000 AMC 8 Problems/Problem 17"
Mrdavid445 (talk | contribs) (Created page with "The operation <math> \otimes </math> is defined for all nonzero numbers by <math> a\otimes b =\frac{a^{2}}{b} </math>. Determine <math> [(1\otimes 2)\otimes 3]-[1\otimes (2\otime...") |
(→Video Solution) |
||
| (4 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
| + | |||
The operation <math> \otimes </math> is defined for all nonzero numbers by <math> a\otimes b =\frac{a^{2}}{b} </math>. Determine <math> [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] </math>. | The operation <math> \otimes </math> is defined for all nonzero numbers by <math> a\otimes b =\frac{a^{2}}{b} </math>. Determine <math> [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] </math>. | ||
<math> \text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3} </math> | <math> \text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3} </math> | ||
| + | |||
| + | ==Solution== | ||
| + | Follow PE(MD)(AS), doing the innermost parentheses first. | ||
| + | |||
| + | <math> [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] </math> | ||
| + | |||
| + | <math> [\frac{1^2}{2}\otimes 3]-[1\otimes \frac{2^2}{3}] </math> | ||
| + | |||
| + | <math> [\frac{1}{2}\otimes 3]-[1\otimes \frac{4}{3}] </math> | ||
| + | |||
| + | <math> [\frac{(\frac{1}{2})^{2}}{3}]-[\frac{1^2}{(\frac{4}{3})}] </math> | ||
| + | |||
| + | <math> [\frac{1}{4} \cdot \frac{1}{3}]-[\frac{3}{4}] </math> | ||
| + | |||
| + | <math>\frac{1}{12} - \frac{3}{4}</math> | ||
| + | |||
| + | <math>\frac{1}{12} - \frac{9}{12}</math> | ||
| + | |||
| + | <math>\frac{-8}{12}</math> | ||
| + | |||
| + | <math>-\frac{2}{3}</math>, which is answer <math>\boxed{A}</math> | ||
| + | |||
| + | |||
| + | == Video Solution == | ||
| + | |||
| + | https://www.youtube.com/watch?v=dJmKnoKqjX4 ~David | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{AMC8 box|year=2000|num-b=16|num-a=18}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:05, 15 April 2023
Contents
Problem
The operation 
is defined for all nonzero numbers by 
. Determine ![$[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$](http://latex.artofproblemsolving.com/f/9/c/f9cbc296f6d18eac4c7b30b42a83fc69c6e32162.png)
.
Solution
Follow PE(MD)(AS), doing the innermost parentheses first.
![$[\frac{1^2}{2}\otimes 3]-[1\otimes \frac{2^2}{3}]$](http://latex.artofproblemsolving.com/1/c/3/1c3b42cf303113f7704e69ad5ed83130da3e39ff.png)
![$[\frac{1}{2}\otimes 3]-[1\otimes \frac{4}{3}]$](http://latex.artofproblemsolving.com/9/a/b/9abcf9d98488337d87742f88597a37a35779e639.png)
![$[\frac{(\frac{1}{2})^{2}}{3}]-[\frac{1^2}{(\frac{4}{3})}]$](http://latex.artofproblemsolving.com/d/6/b/d6bb554c3c21ca52249177ed3a59ff64a167f3ff.png)
![$[\frac{1}{4} \cdot \frac{1}{3}]-[\frac{3}{4}]$](http://latex.artofproblemsolving.com/4/7/3/47306518909ceea5ecfd7e3f14b6b81726c8ec6e.png)
, which is answer 
Video Solution
https://www.youtube.com/watch?v=dJmKnoKqjX4 ~David
See Also
| 2000 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
