Difference between revisions of "2016 AMC 10B Problems/Problem 3"
Doitsudoitsu (talk | contribs) m (basic solution; someone should add detailed edits in; this was rushed.) |
(→Video Solution) |
||
(7 intermediate revisions by 5 users not shown) | |||
Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
− | Let <math>x=-2016</math>. What is the value of <math>\ | + | Let <math>x=-2016</math>. What is the value of <math>\Bigg\vert\Big\vert |x|-x\Big\vert-|x|\Bigg\vert-x</math> ? |
<math>\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048</math> | <math>\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | <math> | + | Substituting carefully, <math>\Bigg\vert\Big\vert 2016-(-2016)\Big\vert-2016\Bigg\vert-(-2016)</math> |
− | <math> | + | |
− | <math>| | + | becomes <math>|4032-2016|+2016=2016+2016=4032</math> which is <math>\boxed{\textbf{(D)}}</math>. |
− | <math> | + | |
+ | ==Solution 2== | ||
+ | |||
+ | Solution by e_power_pi_times_i | ||
+ | |||
+ | Substitute <math>-y = x = -2016</math> into the equation. Now, it is <math>\Bigg\vert\Big\vert |y|+y\Big\vert-|y|\Bigg\vert+y</math>. Since <math>y = 2016</math>, it is a positive number, so <math>|y| = y</math>. Now the equation is <math>\Bigg\vert\Big\vert y+y\Big\vert-y\Bigg\vert+y</math>. This further simplifies to <math>2y-y+y = 2y</math>, so the answer is <math>\boxed{\textbf{(D)}\ 4032}</math> | ||
+ | |||
+ | ==Video Solution (CREATIVE THINKING)== | ||
+ | https://youtu.be/2cPRcI_rIvU | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | |||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/PuAY2v0bt-0 | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2016|ab=B|num-b=2|num-a=4}} | ||
+ | {{MAA Notice}} |
Latest revision as of 12:48, 2 July 2023
Contents
Problem
Let . What is the value of
?
Solution 1
Substituting carefully,
becomes which is
.
Solution 2
Solution by e_power_pi_times_i
Substitute into the equation. Now, it is
. Since
, it is a positive number, so
. Now the equation is
. This further simplifies to
, so the answer is
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.