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

Revision as of 17:06, 27 August 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

If $x\neq y,$ then $\frac{x-y}{y-x}=-1.$ We use this fact to simplify the original expression: $\frac{\color{red}\overset{-1}{\cancel{a-3}}}{\color{blue}\underset{1}{\cancel{5-c}}} \cdot \frac{\color{green}\overset{-1}{\cancel{b-4}}}{\color{red}\underset{1}{\cancel{3-a}}} \cdot \frac{\color{blue}\overset{-1}{\cancel{c-5}}}{\color{green}\underset{1}{\cancel{4-b}}}=(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}.$ ~CoolJupiter (Solution)

~MRENTHUSIASM ( $\LaTeX$ Adjustments)

Solution 2

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 3

At $(a,b,c)=(4,5,6),$ the answer choices become

$\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } -\frac{1}{120} \qquad \textbf{(E) } \frac{1}{120}$

and the original expression becomes $$ (Error compiling LaTeX. Unknown error_msg)\frac{-1}{1}\cdot\frac{-1}{1}\cdot\frac{-1}{1}=\boxed{\textbf{(A) } -1}.$

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

@@ Line 6: / Line 6: @@
 == Solution 1 ==
-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>
 ~CoolJupiter (Solution)
@@ Line 13: / Line 13: @@
 == Solution 2 ==
-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>
 == Solution 3 ==
-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>.
+At <math>(a,b,c)=(4,5,6),</math> the answer choices become
-Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A) } -1}</math>.
+<math>\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } -\frac{1}{120} \qquad \textbf{(E) } \frac{1}{120}</math>
+and the original expression becomes <math></math>\frac{-1}{1}\cdot\frac{-1}{1}\cdot\frac{-1}{1}=\boxed{\textbf{(A) } -1}.$
 == Video Solution 1 ==

2020 AMC 10A (Problems • Answer Key • Resources)
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