Competition · AMC preparation · step 4 of 4
AMC 8 · 2017 · #21
Grade 7 algebralogicPick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The scary-looking expression collapses once we solve the easier sub-question x/|x|=? for any nonzero x (Tool #9). Each term is just a sign, +1 or -1. After that, we want to know which sign-patterns of (a,b,c) are even allowed: there are only 2³=8 patterns, so we list them systematically (Tool #2) and use the constraint a+b+c=0 to eliminate the all-positive and all-negative cases (Tool #3). Two sign-pattern families survive, and by symmetry we only have to evaluate the expression once per family.
Simplify each sign fraction
The easier piece x/|x| is +1 when x is positive and -1 when x is negative — so every term is just +1 or -1, the variable's sign.
Absolute value just strips the sign, so dividing a number by its own absolute value leaves only the sign behind.
6.NS.C.7Solve An Easier Related ProblemList all eight sign patterns
List all 8 sign patterns of (a,b,c), ordered by how many are negative, then filter them with the constraint a+b+c=0.
There are only finitely many sign patterns, so we can just walk through them in order without missing any.
6.NS.C.5Make A Systematic ListDrop the impossible patterns
All-positive sums too high and all-negative too low, so both are impossible — leaving exactly 1 or 2 negatives among a, b, c.
A sum of three numbers can only equal zero if they have mixed signs — that's what the constraint a+b+c=0 is telling us.
6.NS.C.5Eliminate PossibilitiesEvaluate the one-negative case
Family 1 (two positives, one negative): the three signs give 1+1-1=1, and abc is negative so its term is -1 — total 0.
The sign of a product is determined by how many factors are negative; here an odd count (one) gives a negative product.
When two of a, b, c are positive and one is negative, the whole expression a/|a|+b/|b|+c/|c|+abc/|abc| equals 0.
▸ Why?
The expression splits into two parts whose values are +1 and -1, and 1+(-1)=0.
▸ Why?
The three single-variable terms a/|a|,b/|b|,c/|c| add to +1, because two of the variables are positive (each term is +1) and one is negative (its term is -1), so 1+1+(-1)=1.
▸ Why?
For a nonzero x, x/|x| is +1 when x ＞ 0 and -1 when x ＜ 0: since x equals its sign times |x|, dividing x by |x| reverses that multiplication and hands back just the sign.
▸ Why?
The fourth term abc/|abc| is -1, because the product abc of two positive factors and one negative factor is negative.
▸ Why?
Multiplying the two positive factors first gives a positive number, and multiplying that positive result by the one negative factor gives a negative number.
▸ Why?
A whole count of equal groups of a positive amount piles up to a positive total, so a positive times a positive is positive.
▸ Why?
A whole count of equal groups of a negative amount piles up to a negative total, so a positive times a negative is negative.
Evaluate the two-negative case
Family 2 (one positive, two negatives): signs give 1-1-1=-1, abc positive so +1 — total 0 again. Same both ways, so answer (A).
Two negatives multiply to a positive, so (+)(-)(-) is positive — and the two cases conspire to give the same total.
7.NS.A.2Solve An Easier Related ProblemThis AMC 8 problem only needs Grade 7 sign-of-a-product reasoning with positive and negative numbers you already know!
- Simplify each sign fraction
- List all eight sign patterns
- Drop the impossible patterns
- Evaluate the one-negative case
- Evaluate the two-negative case
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