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

## 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

~IceMatrix

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