Difference between revisions of "2020 AMC 10A Problems/Problem 3"
MRENTHUSIASM (talk | contribs) (I will clean up this page and consider some of the edits.) |
MRENTHUSIASM (talk | contribs) (Deleted original Sol 2 as it is repetitive. Maintained the solution by answer choices, as that's a last resort. Let me know if you disagree with this edit.) |
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<math>\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}</math> | <math>\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}</math> | ||
| − | == Solution 1 == | + | == Solution 1 (Negatives) == |
If <math>x\neq y,</math> then <math>\frac{x-y}{y-x}=-1.</math> We use this fact to simplify the original expression: | If <math>x\neq y,</math> then <math>\frac{x-y}{y-x}=-1.</math> We use this fact to simplify the original expression: | ||
<cmath>\frac{\color{red}\overset{-1}{\cancel{a-3}}}{\color{blue}\underset{1}{\cancel{5-c}}} \cdot \frac{\color{green}\overset{-1}{\cancel{b-4}}}{\color{red}\underset{1}{\cancel{3-a}}} \cdot \frac{\color{blue}\overset{-1}{\cancel{c-5}}}{\color{green}\underset{1}{\cancel{4-b}}}=(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}.</cmath> | <cmath>\frac{\color{red}\overset{-1}{\cancel{a-3}}}{\color{blue}\underset{1}{\cancel{5-c}}} \cdot \frac{\color{green}\overset{-1}{\cancel{b-4}}}{\color{red}\underset{1}{\cancel{3-a}}} \cdot \frac{\color{blue}\overset{-1}{\cancel{c-5}}}{\color{green}\underset{1}{\cancel{4-b}}}=(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}.</cmath> | ||
| − | ~CoolJupiter | + | ~CoolJupiter ~MRENTHUSIASM |
| − | + | == Solution 2 (Answer Choices) == | |
| − | |||
| − | == Solution 2 | ||
| − | |||
| − | |||
| − | |||
At <math>(a,b,c)=(4,5,6),</math> the answer choices become | At <math>(a,b,c)=(4,5,6),</math> the answer choices become | ||
Revision as of 18:29, 4 November 2021
Contents
Problem
Assuming 
, 
, and 
, what is the value in simplest form of the following expression?
![\[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\]](http://latex.artofproblemsolving.com/9/2/6/926ff7e4cb0ce3392334f8215dd3aa4c51438f15.png)
Solution 1 (Negatives)
If 
then 
We use this fact to simplify the original expression:
![\[\frac{\color{red}\overset{-1}{\cancel{a-3}}}{\color{blue}\underset{1}{\cancel{5-c}}} \cdot \frac{\color{green}\overset{-1}{\cancel{b-4}}}{\color{red}\underset{1}{\cancel{3-a}}} \cdot \frac{\color{blue}\overset{-1}{\cancel{c-5}}}{\color{green}\underset{1}{\cancel{4-b}}}=(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}.\]](http://latex.artofproblemsolving.com/8/3/5/8355b76578ba64de142880ae5e5cb276d768fb35.png)
~CoolJupiter ~MRENTHUSIASM
Solution 2 (Answer Choices)
At 
the answer choices become
and the original expression becomes ![\[\frac{-1}{1}\cdot\frac{-1}{1}\cdot\frac{-1}{1}=\boxed{\textbf{(A) } -1}.\]](http://latex.artofproblemsolving.com/5/1/a/51a53110edb038c9ac16c09b37ed76322eebc55b.png)
~MRENTHUSIASM
Video Solution 1
~IceMatrix
Video Solution 2
Education, The Study of Everything
Video Solution 3
https://www.youtube.com/watch?v=7-3sl1pSojc
~bobthefam
Video Solution 4
~savannahsolver
Video Solution 5
https://youtu.be/ba6w1OhXqOQ?t=956
~ pi_is_3.14
See Also
| 2020 AMC 10A (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. 
