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

Revision as of 13:28, 22 December 2020

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 $a, b,\text{ and } c$ , we see that if each of them satify 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.

$\frac{\cancel{(a-3)} -1 \cancel{(b-4)} -1 \cancel{(c-5)} -1}{\cancel{(5-c)}\cancel{(3-a)}\cancel{(4-b)}}=(-1)(-1)(-1)=\boxed{\textbf{(A)}-1}$

~CoolJupiter

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

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.

@@ Line 9: / Line 9: @@
 == Solution 2 ==
-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>.
+Substituting values for <math>a, b,\text{ and } c</math>, we see that if each of them satify the inequalities above, the value goes to be <math>-1</math>.
 Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A)}-1}</math>.
@@ Line 19: / Line 19: @@
 ~CoolJupiter
-=== Video Solution 1 ==
+== Video Solution 1 ==
 https://youtu.be/WUcbVNy2uv0
 ~IceMatrix
-=== Video Solution 2 ==
+== Video Solution 2 ==
 https://youtu.be/Nrdxe4UAqkA
@@ Line 30: / Line 30: @@
 Education, The Study of Everything
-=== Video Solution 3 ==
+== Video Solution 3 ==
 https://www.youtube.com/watch?v=7-3sl1pSojc
 ~bobthefam
-=== Video Solution 4 ==
+== Video Solution 4 ==
 https://youtu.be/ZccL6yKrTiU

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