2017 AMC 8 Problems/Problem 21

Problem

Suppose $a$ , $b$ , and $c$ are nonzero real numbers, and $a+b+c=0$ . What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$ ?

$\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1$

Solution

There are $2$ cases to consider:

Case $1$ : $2$ of $a$ , $b$ , and $c$ are positive and the other is negative. Without loss of generality (WLOG), we can assume that $a$ and $b$ are positive and $c$ is negative. In this case, we have that $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=1+1-1-1=0.$

Case $2$ : $2$ of $a$ , $b$ , and $c$ are negative and the other is positive. WLOG, we can assume that $a$ and $b$ are negative and $c$ is positive. In this case, we have that $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=-1-1+1+1=0.$

Note these are the only valid cases, for neither $3$ negatives nor $3$ positives would work as they cannot sum up to $0$ . In both cases, we get that the given expression equals $\boxed{\textbf{(A)}\ 0}$ .

Video Solution (CREATIVE THINKING + ANALYSIS!!!)

https://youtu.be/wPVQwzA-2hU

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/7an5wU9Q5hk?t=2362

Video Solutions

https://youtu.be/FUEHirfk-tw

https://youtu.be/V9wCBTwvIZo

- Happytwin

https://youtu.be/xN0dnJC1hv8

~savannahsolver

https://www.youtube.com/watch?v=u6M6ECGok1o

- SUS

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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All AJHSME/AMC 8 Problems and Solutions

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