Competition · AMC preparation · step 4 of 4

AMC 8 · 2000 · #17

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

fraction-arithmeticfunction-evaluationorder-of-operations identify-subproblemsformula-substitution ↑ Prerequisites: fraction-arithmeticorder-of-operations

📏 Medium solution 💡 3 insights

Problem

A new operation is defined by a ⊗ b =

\frac{a²}{b}

for all nonzero numbers. Compute [(1 ⊗ 2) ⊗ 3] - [1 ⊗ (2 ⊗ 3)] — the same three numbers, but the brackets group them differently on the two sides.

Pick an answer.

(A)

$-\frac{2}{3}$

(B)

$-\frac{1}{4}$

(C)

(D)

$\frac{1}{4}$

(E)

$\frac{2}{3}$

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The bracket structure splits the work into four small substitutions — Tool #7 (Identify Subproblems) handles it cleanly. Evaluate the inner ⊗ on each side, then feed the result into the outer ⊗, then subtract. Because ⊗ is non-associative (the left input gets squared, the right one does not), the two sides will land on different numbers; the difference is the answer. Tool #3 (Eliminate Possibilities) is the AMC multiple-choice safety net: a quick sign check (left side is small, right side is close to 1, so the difference must be negative) already rules out (C), (D), (E).

1STEP 1

Split into four substitutions

Split into four substitutions: each side's inner ⊗ first, then its outer ⊗, then subtract left minus right.

[(1 ⊗ 2) ⊗ 3] - [1 ⊗ (2 ⊗ 3)]

2STEP 2

Do the left inner operation

Left inner ⊗: with a = 1, b = 2, square 1 and divide by 2 to get $\frac{1}{2}$ .

1 ⊗ 2 = 1²/2 = 1/2

3STEP 3

Do the left outer operation

Left outer ⊗: feed $\frac{1}{2}$ in as a with b = 3, square to $\frac{1}{4}$ , then divide by 3 to get $\frac{1}{12}$ .

(1/2) ⊗ 3 = (1/2)²/3 = 1/4/3 = 1/12

4STEP 4

Do the right inner operation

Right inner ⊗: with a = 2, b = 3, square 2 to get 4, then divide by 3, giving $\frac{4}{3}$ .

2 ⊗ 3 = 2²/3 = 4/3

5STEP 5

Do the right outer operation

Right outer ⊗: square 1, then divide by $\frac{4}{3}$ — the same as multiplying by its reciprocal $\frac{3}{4}$ — giving $\frac{3}{4}$ .

1 ⊗ 4/3 = 1²/4/3 = 1 · 3/4 = 3/4

6STEP 6

Subtract the two values

Subtract left minus right over common denominator 12: $\frac{1}{12}$ - $\frac{9}{12}$ = $-\frac{2}{3}$ .

1/12 - 3/4 = 1/12 - 9/12 = -8/12 = -2/3

Rewriting 3/4 as 9/12 is the Grade 5 "common denominator" step, then simplify -8/12 by dividing top and bottom by 4.

The whole expression [(1 ⊗ 2) ⊗ 3] − [1 ⊗ (2 ⊗ 3)] comes out to −2/3.

▸ Why?

The left grouping first turns 1 ⊗ 2 into 1/2, then applies ⊗ 3 to that, and squaring the fraction 1/2 shrinks the left side to (1/2)²/3 = 1/12.

▸ Why?

By the rule, 1 ⊗ 2 squares the left input 1 and divides by 2, and squaring 1 leaves it unchanged, so 1 ⊗ 2 = 1/2.

Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.

▸ Why?

Squaring the fraction 1/2 means multiplying it by itself, which gives 1/4.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

Dividing 1/4 by 3 asks which number tripled returns 1/4, and that is 1/12, since 1/12 taken 3 times is 1/4.

Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

▸ Why?

The right grouping first turns 2 ⊗ 3 into 4/3, then applies 1 ⊗ to that, and squaring the left input 1 keeps it at 1, so the right side stays near one: 1 ⊗ (4/3) = 1/4/3 = 3/4.

▸ Why?

By the rule, 2 ⊗ 3 squares the left input 2 to get 4 and divides by 3, so 2 ⊗ 3 = 4/3; squaring 2 is 2 groups of 2, which is 4.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

Squaring the left input 1 leaves it as 1, so 1 ⊗ (4/3) becomes just 1 divided by 4/3.

Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.

▸ Why?

Dividing 1 by 4/3 asks which number times 4/3 gives 1, and that is its reciprocal 3/4.

Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

▸ Why?

Because the left side 1/12 is far smaller than the right side 3/4, taking the right away from the left drops below zero to −2/3.

▸ Why?

Rename 3/4 with denominator 12 by multiplying its top and bottom both by 3, which keeps the same value, giving 3/4 = 9/12.

Equivalent fractionsMultiplying (or dividing) a fraction's top and bottom by the same nonzero number leaves its value unchanged.

▸ Why?

With both written as twelfths, 1/12 minus 9/12 is 1 twelfth take away 9 twelfths, which is (1 − 9) twelfths = −8/12.

Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.

▸ Why?

Simplify −8/12 by dividing its top and bottom both by 4, which keeps the same value, giving −2/3.

Equivalent fractionsMultiplying (or dividing) a fraction's top and bottom by the same nonzero number leaves its value unchanged.

5.NF.A.1Identify Subproblems

7STEP 7

Match against the choices

Match $-\frac{2}{3}$ to the choices: only (A) fits; a quick sign check (right $\frac{3}{4}$ beats left $\frac{1}{12}$ ) already kills (C), (D), (E).

-2/3 → (A)

Answer

-2/3

Sanity check the sign and size. The right-side answer

\frac{3}{4}

is close to 1 because dividing 1 by something a little bigger than 1 (namely

\frac{4}{3}

) gives a number a little smaller than 1. The left-side answer

\frac{1}{12}

is much smaller because we squared a fraction (

\frac{1}{2}

→

\frac{1}{4}

) and then divided by another whole number. So left - right is small minus near-1 — clearly negative and not tiny, which matches

-\frac{2}{3}

. The non-zero answer also confirms that ⊗ is not associative; if it were, the difference would be 0 (choice (C)), so the problem is really testing that observation.

💡Key takeaway

Brackets pick which number gets squared, and squaring a fraction shrinks it while squaring a whole number grows it — that asymmetry is exactly why the two sides do not match, and the difference comes out to $-\frac{2}{3}$ , answer (A).

Split into four substitutions
Do the left inner operation
Do the left outer operation
Do the right inner operation
Do the right outer operation
Subtract the two values
Match against the choices

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Split into four substitutions

Do the left inner operation

Do the left outer operation

Do the right inner operation

Do the right outer operation

Subtract the two values

Match against the choices

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