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

Revision as of 20:42, 24 May 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

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 $(-1)(-1)(-1)=\boxed{\textbf{(A)}-1}$ .

Video Solution

https://youtu.be/WUcbVNy2uv0

~IceMatrix

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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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, 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 <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>.
 Therefore, the product of the given fraction equals <math>(-1)(-1)(-1)=\boxed{\textbf{(A)}-1}</math>.

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