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

Revision as of 15:34, 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

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

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)

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 13: / Line 13: @@
 == Solution 3 ==
-Note that <math>\frac{x-y}{y-x}=-1</math> for <math>x\ne y.</math> We use this fact to cancel out the factors: <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>
+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)

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