Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #7
Grade 7 arithmeticPick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We want the most negative product, so Tool #14 (Extreme Principle) applies: the answer comes from pushing the chosen numbers to extreme magnitudes with the right signs. The sign rules narrow the search: three numbers multiply to a negative only when the count of negatives is 1 or 3, so Tool #2 (List Out Cases) gives just two cases to compare. Inside each case, the Extreme Principle picks the numbers with the largest absolute values. Skip 0 — any product containing it is 0, which is not the smallest.
List the sign patterns
The product is negative only when an odd number of factors are negative — 1 or 3 negatives; any pick with 0 gives 0, so drop 0.
The Grade 7 sign rule for multiplication says each negative factor flips the sign once. An odd number of flips leaves the product negative.
7.NS.A.2Make A Systematic ListMultiply the three negatives
Case A takes all three negatives -8, -6, -4: two negatives make a positive, the third flips it back to -192.
No choice to make in this case — the three negatives are fixed, so the product is forced to -192.
7.NS.A.2Make A Systematic ListPick the one-negative trio
Case B (one negative, two positives): the Extreme Principle takes the biggest-size negative and two biggest positives — -8, 7, 5.
Bigger magnitudes on each factor make the size of the product bigger. Since the sign is locked negative by the one negative factor, bigger size means more negative.
In the case that uses one negative number and two positive numbers, the product is pushed as negative as it can go by choosing the negative number farthest from zero, -8, together with the two largest positive numbers, 7 and 5.
▸ Why?
With one negative factor and the other two positive, the product is a negative amount, so making it "as negative as possible" means making that amount reach as far below zero as it can.
▸ Why?
Multiplying the lone negative by the positive numbers just adds that negative amount together in equal groups, and adding a negative amount over and over keeps the running total negative.
▸ Why?
A negative number sits below zero, and the farther below zero it lies the smaller it is, so pushing the amount farther from zero makes its value smaller.
▸ Why?
How far the product lands from zero is the three sizes multiplied, 8 × 7 × 5, and that distance grows whenever one factor is swapped for a larger one, so we take the biggest size in each slot: 8 from the negatives and 7 and 5 from the positives.
▸ Why?
A product is just that many equal groups of a size, so trading a factor for a larger one gives either more groups or bigger groups, and either way the total adds up to more.
Compare the two cases
Case B gives (-8)×7×5 = -280; since -280 ＜ -192 on the number line, the minimum is -280, choice (B).
Comparing two negative numbers: the one farther from 0 is smaller. -280 is farther left on the number line than -192.
6.NS.C.7Extreme PrincipleFor a smallest-product question, list only the sign patterns that go negative, then within each one make every factor as big in size as possible. Two short cases beat any guess-and-shuffle approach.
- List the sign patterns
- Multiply the three negatives
- Pick the one-negative trio
- Compare the two cases
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