Difference between revisions of "2022 AMC 8 Problems/Problem 8"
MRENTHUSIASM (talk | contribs) (→Solution) |
(→Solution) |
||
Line 5: | Line 5: | ||
==Solution== | ==Solution== | ||
+ | The key observation in this problem is that all of the numbers from <math>3</math> to <math>20</math> (inclusive) cancel out. To see this, notice that for all numbers <math>3</math> to <math>20</math> (inclusive) there's one of them in a numerator and another in a denominator. Thus, the entire expression simplifies to | ||
+ | <cmath>\frac{1}{\cancel{3}}\cdot\frac{2}{\cancel{4}}\cdot\frac{\cancel{3}}{\cancel{5}}\cdots\frac{\cancel{18}}{\cancel{20}}\cdot\frac{\cancel{19}}{21}\cdot\frac{\cancel{20}}{22} = \frac{1\cdot 2}{21 \cdot 22} = \frac{1}{21\cdot 11}.</cmath> | ||
+ | |||
+ | Thus, our answer is <math>\boxed{\textbf{(B) } \frac{1}{231}}</math> | ||
Note that common factors of the numerator and the denominator cancel, so the original expression becomes <cmath>\frac{1}{\cancel{3}}\cdot\frac{2}{\cancel{4}}\cdot\frac{\cancel{3}}{\cancel{5}}\cdots\frac{\cancel{18}}{\cancel{20}}\cdot\frac{\cancel{19}}{21}\cdot\frac{\cancel{20}}{22}=\frac{1\cdot2}{21\cdot22}=\frac{1}{21\cdot11}=\boxed{\textbf{(B) } \frac{1}{231}}.</cmath> | Note that common factors of the numerator and the denominator cancel, so the original expression becomes <cmath>\frac{1}{\cancel{3}}\cdot\frac{2}{\cancel{4}}\cdot\frac{\cancel{3}}{\cancel{5}}\cdots\frac{\cancel{18}}{\cancel{20}}\cdot\frac{\cancel{19}}{21}\cdot\frac{\cancel{20}}{22}=\frac{1\cdot2}{21\cdot22}=\frac{1}{21\cdot11}=\boxed{\textbf{(B) } \frac{1}{231}}.</cmath> |
Revision as of 11:00, 28 January 2022
Problem
What is the value of
Solution
The key observation in this problem is that all of the numbers from to (inclusive) cancel out. To see this, notice that for all numbers to (inclusive) there's one of them in a numerator and another in a denominator. Thus, the entire expression simplifies to
Thus, our answer is
Note that common factors of the numerator and the denominator cancel, so the original expression becomes
~MRENTHUSIASM
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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.