Difference between revisions of "2020 AMC 10A Problems/Problem 3"

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==Problem==
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== Problem ==
 
Assuming <math>a\neq3</math>, <math>b\neq4</math>, and <math>c\neq5</math>, what is the value in simplest form of the following expression?
 
Assuming <math>a\neq3</math>, <math>b\neq4</math>, and <math>c\neq5</math>, what is the value in simplest form of the following expression?
 
<cmath>\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}</cmath>
 
<cmath>\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}</cmath>
 +
 
<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 ==  
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== Solution 1 ==
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Note that <math>a-3</math> is <math>-1</math> times <math>3-a</math>. Likewise, <math>b-4</math> is <math>-1</math> times <math>4-b</math> and <math>c-5</math> is <math>-1</math> times <math>5-c</math>. Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}</math>.
  
Note that <math>a-3</math> is <math>-1</math> times <math>3-a</math>. Likewise, <math>b-4</math> is <math>-1</math> times <math>4-b</math> and <math>c-5</math> is <math>-1</math> times <math>5-c</math>. Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A)}-1}</math>.
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== Solution 2 ==
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Substituting values for <math>a, b,</math> and <math>c</math>, we see that if each of them satisfy the inequalities above, the value goes to be <math>-1</math>.  
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Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}</math>.
  
==Solution 2==
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== Solution 3 ==
Substituting values for <cmath>a, b,\text{and} c</cmath>, we see that if each of them satify the inequalities above, the value goes to be <cmath>-1</cmath>.
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It is known that <math>\frac{x-y}{y-x}=-1</math> for <math>x\ne y.</math> We use this fact to cancel out the terms: <cmath>\frac{\overset{-1}{\cancel{a-3}}}{\underset{1}{\xcancel{5-c}}} \cdot \frac{\overset{-1}{\bcancel{b-4}}}{\underset{1}{\cancel{3-a}}} \cdot \frac{\overset{-1}{\xcancel{c-5}}}{\underset{1}{\bcancel{4-b}}}=(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}.</cmath>
Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A)}-1}</math>.
 
  
==Solution 3==
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~CoolJupiter (Solution)
It is known that <math>\frac{x-y}{y-x}=-1</math> for <math>x\ne y</math>. We use this fact to cancel out the terms.
 
  
<math>\frac{\cancel{(a-3)} -1 \cancel{(b-4)} -1 \cancel{(c-5)} -1}{\cancel{(5-c)}\cancel{(3-a)}\cancel{(4-b)}}=(-1)(-1)(-1)=\boxed{\textbf{(A)}~-1}</math>  
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~MRENTHUSIASM (<math>\LaTeX</math> Adjustments)
  
~CoolJupiter
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== Video Solution 1 ==
 
 
==Video Solution==
 
 
https://youtu.be/WUcbVNy2uv0
 
https://youtu.be/WUcbVNy2uv0
  
 
~IceMatrix
 
~IceMatrix
  
 +
== Video Solution 2 ==
 +
 +
https://youtu.be/Nrdxe4UAqkA
 +
 +
Education, The Study of Everything
 +
 +
== Video Solution 3 ==
 
https://www.youtube.com/watch?v=7-3sl1pSojc
 
https://www.youtube.com/watch?v=7-3sl1pSojc
  
 
~bobthefam
 
~bobthefam
  
 +
== Video Solution 4 ==
 
https://youtu.be/ZccL6yKrTiU
 
https://youtu.be/ZccL6yKrTiU
  
 
~savannahsolver
 
~savannahsolver
  
==See Also==
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== Video Solution 5==
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https://youtu.be/ba6w1OhXqOQ?t=956
 +
 
 +
~ pi_is_3.14
  
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== See Also ==
 
{{AMC10 box|year=2020|ab=A|num-b=2|num-a=4}}
 
{{AMC10 box|year=2020|ab=A|num-b=2|num-a=4}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 16:32, 29 April 2021

Problem

Assuming $a\neq3$, $b\neq4$, and $c\neq5$, what is the value in simplest form of the following expression? \[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\]

$\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}$

Solution 1

Note that $a-3$ is $-1$ times $3-a$. Likewise, $b-4$ is $-1$ times $4-b$ and $c-5$ is $-1$ times $5-c$. Therefore, the product of the given fraction equals $(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}$.

Solution 2

Substituting values for $a, b,$ and $c$, we see that if each of them satisfy the inequalities above, the value goes to be $-1$. Therefore, the product of the given fraction equals $(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}$.

Solution 3

It is known that $\frac{x-y}{y-x}=-1$ for $x\ne y.$ We use this fact to cancel out the terms: \[\frac{\overset{-1}{\cancel{a-3}}}{\underset{1}{\xcancel{5-c}}} \cdot \frac{\overset{-1}{\bcancel{b-4}}}{\underset{1}{\cancel{3-a}}} \cdot \frac{\overset{-1}{\xcancel{c-5}}}{\underset{1}{\bcancel{4-b}}}=(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}.\]

~CoolJupiter (Solution)

~MRENTHUSIASM ($\LaTeX$ Adjustments)

Video Solution 1

https://youtu.be/WUcbVNy2uv0

~IceMatrix

Video Solution 2

https://youtu.be/Nrdxe4UAqkA

Education, The Study of Everything

Video Solution 3

https://www.youtube.com/watch?v=7-3sl1pSojc

~bobthefam

Video Solution 4

https://youtu.be/ZccL6yKrTiU

~savannahsolver

Video Solution 5

https://youtu.be/ba6w1OhXqOQ?t=956

~ pi_is_3.14

See Also

2020 AMC 10A (ProblemsAnswer KeyResources)
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Problem 2
Followed by
Problem 4
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All AMC 10 Problems and Solutions

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