AMC 10 · 2010 · #11

Grade 7 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

percentagelinear-equations-one-varinterval-arithmetic convert-to-algebra ↑ Prerequisites: percentage

📏 Medium solution 💡 3 insights

Problem

A shopper plans to purchase an item that has a listed price greater than $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\textdollar 100$ .
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$ ?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

100

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: An item is listed at a price above $\textdollar 100$. Coupon A takes $15\%$ off the full price, Coupon B takes $\textdollar 30$ off, and Coupon C takes $25\%$ off only the part of the price above $\textdollar 100$. Let $x$ be the smallest price and $y$ the largest price at which Coupon A saves at least as much as both Coupon B and Coupon C. Find $y-x$.

Givens: The listed price is greater than $\textdollar 100$; Coupon A saves $15\%$ of the listed price; Coupon B saves a flat $\textdollar 30$; Coupon C saves $25\%$ of the amount above $\textdollar 100$; Answer choices: (A) $50$, (B) $60$, (C) $75$, (D) $80$, (E) $100$

Unknowns: The smallest price $x$ and largest price $y$ where Coupon A is at least as good as both B and C; The difference $y-x$

Understand

Plan

Primary tool: #4 Introduce a Variable

Secondary: #13 Convert to Algebra, #14 Extreme Principle

The price is not given, so Tool #4 (Introduce a Variable) names it $P$ and turns each coupon's savings into an expression in $P$. Tool #13 (Convert to Algebra) then rewrites the words "A saves at least as much as B" and "A saves at least as much as C" as two inequalities. Solving them pins down a range of allowed prices, and Tool #14 (Extreme Principle) reads off the two endpoints of that range as the smallest price $x$ and largest price $y$.

Execute — Answer: A

#4 Introduce a Variable 7.RP.A.3 Step 1

Name the price, write each saving

Let the listed price be $P$ dollars, with $P>100$.
Coupon A saves $15\%$ of $P$, which is $0.15P$.
Coupon B saves a flat $\textdollar 30$.
Coupon C saves $25\%$ of the part above $\textdollar 100$, which is $0.25(P-100)$.
Writing all three as dollar amounts lets us compare them directly.

$$A=0.15P,\quad B=30,\quad C=0.25(P-100)$$

💡 Turn each coupon's rule into a dollar formula so they can be compared on the same scale.

#13 Convert to Algebra 7.EE.B.4 Step 2

A at least as good as B

We need Coupon A to save at least as many dollars as Coupon B: $0.15P\ge 30$.
Dividing both sides by $0.15$ gives $P\ge 200$.
So the price must be at least $\textdollar 200$ for A to match or beat B.

$$0.15P\ge 30 \;\Rightarrow\; P\ge \frac{30}{0.15}=200$$

💡 A percentage-off deal only overtakes a fixed dollar discount once the price is large enough.

#13 Convert to Algebra 7.EE.B.4 Step 3

A at least as good as C

We also need Coupon A to save at least as many dollars as Coupon C: $0.15P\ge 0.25(P-100)$.
Expanding the right side gives $0.15P\ge 0.25P-25$.
Subtract $0.15P$ and add $25$: $25\ge 0.10P$, so $P\le 250$.
The price must be at most $\textdollar 250$ for A to match or beat C.

$$0.15P\ge 0.25(P-100)\;\Rightarrow\;25\ge 0.10P\;\Rightarrow\;P\le 250$$

💡 C only discounts the part above $\textdollar 100$, so it overtakes A once the price climbs high enough.

#14 Extreme Principle 4.OA.A.3 Step 4

Read the endpoints and subtract

Both conditions must hold at once, so $200\le P\le 250$.
The smallest such price is $x=200$ and the largest is $y=250$.
The difference is $y-x=250-200=50$.
The answer is (A).

$$200\le P\le 250\;\Rightarrow\;x=200,\;y=250,\;y-x=50\;\Rightarrow\;\textbf{(A)}$$

💡 Two bounds squeeze the price into a band; its two ends are exactly the smallest and largest prices asked for.

[1] #4 7.RP.A.3 Let the listed price be $P$ dollars, with $P>100$. Coupon A saves $15\%$ of $P$,

[2] #13 7.EE.B.4 We need Coupon A to save at least as many dollars as Coupon B: $0.15P\ge 30$. Di

[3] #13 7.EE.B.4 We also need Coupon A to save at least as many dollars as Coupon C: $0.15P\ge 0.

[4] #14 4.OA.A.3 Both conditions must hold at once, so $200\le P\le 250$. The smallest such price

Review

Reasonableness: Check the endpoints. At $P=200$: A saves $0.15\times 200=30$, tying B's $\textdollar 30$, and C saves $0.25\times 100=25<30$, so A wins — valid. At $P=250$: A saves $0.15\times 250=37.5$, and C saves $0.25\times 150=37.5$, a tie, while B is only $\textdollar 30$ — valid. Just outside the band A loses: at $P=199$, $A=29.85<30=B$; at $P=251$, $A=37.65<37.75=C$. So $x=200$ and $y=250$ are genuine boundaries and $y-x=50$.

Alternative: Graph the three savings against $P$. Coupon A is a line through the origin with slope $0.15$; Coupon B is a flat line at $30$; Coupon C is a line of slope $0.25$ starting at $P=100$. A sits above B from where they cross ($P=200$) onward, and A sits above C until they cross ($P=250$). The overlap where A is on top of both is $200$ to $250$, giving the same width $50$.

CCSS standards used (min grade 7)

7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems (Converting the $15\%$ and $25\%$ discounts into the dollar expressions $0.15P$ and $0.25(P-100)$.)
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities (Setting up and solving $0.15P\ge 30$ and $0.15P\ge 0.25(P-100)$ to get $P\ge 200$ and $P\le 250$.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Taking the difference of the endpoints, $250-200=50$.)

⭐ Turn each coupon into a dollar formula, write "A beats the others" as two inequalities, and the price range you get has the smallest and largest prices sitting at its two ends.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Name the price, write each saving

A at least as good as B

A at least as good as C

Read the endpoints and subtract

Review

CCSS standards used (min grade 7)

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