AMC 8 · 2005 · #11

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

percentagefraction-multiplicationpattern-recognition identify-subproblemspattern-recognition ↑ Prerequisites: percentage

📏 Short solution 💡 2 insights

Problem

The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up$ 90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?

Pick an answer.

(A)

- extdollar 1.06

(B)

- extdollar 0.53

(C)

extdollar 0

(D)

extdollar 0.53

(E)

extdollar 1.06

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Toolkit + CCSS Solution

Understand

Restated: A $\textdollar 90.00$ coat is on sale with a $20\%$ discount, and sales tax in Rubenenkoville is $6\%$. Jack computes the bill by adding $6\%$ tax to $\textdollar 90$ first and then taking $20\%$ off. Jill takes $20\%$ off $\textdollar 90$ first and then adds $6\%$ tax to the discounted price. What is Jack's total minus Jill's total?

Givens: Original price: $\textdollar 90.00$; Discount: $20\%$ off (multiplier $1 - 0.20 = 0.80$); Sales tax: $6\%$ added (multiplier $1 + 0.06 = 1.06$); Jack's order of operations: price $\to$ add tax $\to$ take discount; Jill's order of operations: price $\to$ take discount $\to$ add tax; Answer choices: (A) $-\textdollar 1.06$, (B) $-\textdollar 0.53$, (C) $\textdollar 0$, (D) $\textdollar 0.53$, (E) $\textdollar 1.06$

Unknowns: The value of Jack's total minus Jill's total, in dollars

Understand

Plan

Primary tool: #11 Find an Invariant

Secondary: #4 Introduce a Variable

The two clerks apply the same two multipliers ($1.06$ for tax and $0.80$ for discount) to the same starting price; only the order is swapped. Tool #11 (Find an Invariant) spots the unchanging quantity: when you multiply a string of numbers, reordering the factors never changes the product. That property — commutativity of multiplication — is the invariant, so Jack's total and Jill's total must be equal before we ever compute a dollar amount. Tool #4 (Introduce a Variable) lets us write both totals as products of $90$, $1.06$, and $0.80$ in two orders so the invariance is visible at a glance. No arithmetic is required to pick the answer; the structure decides it.

Execute — Answer: C

#4 Introduce a Variable 6.RP.A.3 Step 1

Translate each percent change into a single multiplier.
Adding $6\%$ tax to an amount $x$ gives $x + 0.06x = 1.06x$, so "add tax" means $\times 1.06$.
Taking $20\%$ off $x$ gives $x - 0.20x = 0.80x$, so "take discount" means $\times 0.80$.

$$\text{add tax} \equiv \times 1.06, \quad \text{take discount} \equiv \times 0.80$$

💡 A percent change is just one multiplication — turning each step into a clean factor is what makes the order question tractable.

#4 Introduce a Variable 6.EE.A.2 Step 2

Write Jack's total as a product.
Jack starts with $\textdollar 90$, multiplies by $1.06$ (add tax), then multiplies by $0.80$ (take discount).

$$J_{\text{Jack}} = 90 \times 1.06 \times 0.80$$

💡 Each operation in order becomes one factor in a single product expression.

#4 Introduce a Variable 6.EE.A.2 Step 3

Write Jill's total the same way.
Jill starts with $\textdollar 90$, multiplies by $0.80$ (take discount), then multiplies by $1.06$ (add tax).

$$J_{\text{Jill}} = 90 \times 0.80 \times 1.06$$

💡 Same starting price, same two factors — only the order of the last two has changed.

#11 Find an Invariant 6.EE.A.3 Step 4

Apply the invariant.
Multiplication is commutative, so $1.06 \times 0.80 = 0.80 \times 1.06$.
Jack's product and Jill's product therefore use the same three numbers and are equal.
The difference is zero.

$$J_{\text{Jack}} - J_{\text{Jill}} = 90 \times 1.06 \times 0.80 - 90 \times 0.80 \times 1.06 = 0 \;\Rightarrow\; \textbf{(C)}$$

💡 Once both totals are written as the same product of $90$, $1.06$, and $0.80$, the commutative property makes them identical without computing the dollar amount.

[1] #4 6.RP.A.3 Translate each percent change into a single multiplier. Adding $6\%$ tax to an a

[2] #4 6.EE.A.2 Write Jack's total as a product. Jack starts with $\textdollar 90$, multiplies b

[3] #4 6.EE.A.2 Write Jill's total the same way. Jill starts with $\textdollar 90$, multiplies b

[4] #11 6.EE.A.3 Apply the invariant. Multiplication is commutative, so $1.06 \times 0.80 = 0.80

Review

Reasonableness: Compute both totals to confirm. $90 \times 1.06 = 95.40$, then $95.40 \times 0.80 = 76.32$, so Jack's total is $\textdollar 76.32$. $90 \times 0.80 = 72.00$, then $72.00 \times 1.06 = 76.32$, so Jill's total is also $\textdollar 76.32$. Difference: $\textdollar 0$, matching choice (C). The trap answers $\pm\textdollar 0.53$ and $\pm\textdollar 1.06$ would tempt a solver who imagines that taxing first and then discounting gives the tax a "head start" — but multiplying by $1.06$ and $0.80$ in either order produces the same scaling, so no head start exists.

Alternative: Tool #9 (Try a Simpler Case): drop the dollar amount and try a $\textdollar 100$ coat with a $50\%$ discount and $10\%$ tax. Jack: $100 \times 1.10 \times 0.50 = 55$. Jill: $100 \times 0.50 \times 1.10 = 55$. The two routes match for these simpler percents too, hinting that the equality is structural, not numerical — which confirms the commutativity argument for the original numbers.

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world problems, including percent (Translating "add $6\%$ tax" into $\times 1.06$ and "take $20\%$ off" into $\times 0.80$ so each step becomes one multiplication.)
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Writing Jack's and Jill's totals as the products $90 \times 1.06 \times 0.80$ and $90 \times 0.80 \times 1.06$ so they can be compared algebraically.)
6.EE.A.3 Apply the properties of operations to generate equivalent expressions (Using the commutative property of multiplication to recognize $90 \times 1.06 \times 0.80 = 90 \times 0.80 \times 1.06$, so the difference is $0$.)

⭐ Both clerks multiply $\textdollar 90$ by the same two factors — $1.06$ for tax and $0.80$ for discount. The commutative property of multiplication says order does not matter, so the totals are equal and the difference is $\textdollar 0$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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