AMC 10 · 2014 · #11
Grade 7 arithmeticA customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: 10% off the listed price if the listed price is at least \textdollar50
Coupon 2: \textdollar20 off the listed price if the listed price is at least \textdollar100
Coupon 3: 18% off the amount by which the listed price exceeds \textdollar100
For which of the following listed prices will coupon 1 offer a greater price reduction than either coupon 2 or coupon 3?
Pick an answer.
AMC 10 2014 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.
Toolkit + CCSS Solution
Understand
Restated: An appliance has a listed price $P$. The buyer may use exactly one of three coupons: Coupon 1 takes $10\%$ off (if $P \ge \textdollar 50$), Coupon 2 takes $\textdollar 20$ off (if $P \ge \textdollar 100$), Coupon 3 takes $18\%$ off only the part of $P$ above $\textdollar 100$. For which listed price does Coupon 1 give a bigger reduction than both Coupon 2 and Coupon 3?
Givens: Coupon 1 reduction is $10\%$ of the listed price; Coupon 2 reduction is a flat $\textdollar 20$; Coupon 3 reduction is $18\%$ of the amount above $\textdollar 100$; Answer choices: (A) $\textdollar 179.95$, (B) $\textdollar 199.95$, (C) $\textdollar 219.95$, (D) $\textdollar 239.95$, (E) $\textdollar 259.95$; Every choice is above $\textdollar 100$, so all three coupons are allowed
Unknowns: Which listed price makes Coupon 1's reduction larger than both Coupon 2's and Coupon 3's
Understand
Restated: An appliance has a listed price $P$. The buyer may use exactly one of three coupons: Coupon 1 takes $10\%$ off (if $P \ge \textdollar 50$), Coupon 2 takes $\textdollar 20$ off (if $P \ge \textdollar 100$), Coupon 3 takes $18\%$ off only the part of $P$ above $\textdollar 100$. For which listed price does Coupon 1 give a bigger reduction than both Coupon 2 and Coupon 3?
Givens: Coupon 1 reduction is $10\%$ of the listed price; Coupon 2 reduction is a flat $\textdollar 20$; Coupon 3 reduction is $18\%$ of the amount above $\textdollar 100$; Answer choices: (A) $\textdollar 179.95$, (B) $\textdollar 199.95$, (C) $\textdollar 219.95$, (D) $\textdollar 239.95$, (E) $\textdollar 259.95$; Every choice is above $\textdollar 100$, so all three coupons are allowed
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
Let the listed price be $P$ (Tool #4). One variable lets us write all three reductions as expressions instead of testing five prices by hand five times over. Then Tool #13 turns the phrase "Coupon 1 gives a greater reduction" into two inequalities — Coupon 1 vs Coupon 2, and Coupon 1 vs Coupon 3 — that together pin $P$ to a band. Finally Tool #3 checks which of the five listed prices lands inside that band, since only a multiple-choice value can be the answer.
Execute — Answer: C
7.RP.A.3 Step 1 Write the three reductions
- Let $P$ be the listed price.
- Coupon 1 removes $10\%$ of $P$, which is $0.10P$.
- Coupon 2 removes a flat $\textdollar 20$.
- Coupon 3 removes $18\%$ of the part above $\textdollar 100$, which is $0.18(P-100)$.
- Now the three offers are three expressions in the single unknown $P$.
💡 Naming the price $P$ once lets every coupon speak the same language, so they can be compared directly.
7.EE.B.4 Step 2 Coupon 1 must beat Coupon 2
- Require Coupon 1's reduction to be larger than Coupon 2's: $0.10P > 20$.
- Dividing both sides by $0.10$ gives $P > 200$.
- So the listed price has to be above $\textdollar 200$ for the percentage to top the flat $\textdollar 20$.
💡 A fixed $\textdollar 20$ only loses to $10\%$ once the price climbs past $\textdollar 200$.
7.EE.B.4 Step 3 Coupon 1 must beat Coupon 3
- Require Coupon 1's reduction to be larger than Coupon 3's: $0.10P > 0.18(P-100)$.
- Expand the right side: $0.10P > 0.18P - 18$.
- Subtract $0.10P$ from both sides: $0 > 0.08P - 18$, so $0.08P < 18$, giving $P < 225$.
💡 Coupon 3 grows faster than Coupon 1 as the price rises, so Coupon 1 only wins while the price stays below $\textdollar 225$.
6.EE.B.5 Step 4 Find the price in the band
- Both conditions must hold at once, so $200 < P < 225$.
- Check the choices: $\textdollar 179.95$ and $\textdollar 199.95$ are below $200$; $\textdollar 239.95$ and $\textdollar 259.95$ are above $225$; only $\textdollar 219.95$ falls strictly between $200$ and $225$.
- So the answer is (C).
💡 The two inequalities carve out one narrow window, and just one listed price sits inside it.
7.RP.A.3 Let $P$ be the listed price. Coupon 1 removes $10\%$ of $P$, which is $0.10P$. C 7.EE.B.4 Require Coupon 1's reduction to be larger than Coupon 2's: $0.10P > 20$. Dividin 7.EE.B.4 Require Coupon 1's reduction to be larger than Coupon 3's: $0.10P > 0.18(P-100)$ 6.EE.B.5 Both conditions must hold at once, so $200 < P < 225$. Check the choices: $\text Review
Reasonableness: Test $P = 219.95$ directly: Coupon 1 saves $0.10 \cdot 219.95 = \textdollar 21.995$; Coupon 2 saves $\textdollar 20$; Coupon 3 saves $0.18 \cdot 119.95 \approx \textdollar 21.59$. Coupon 1's $\textdollar 22.00$ is the largest of the three, so it does beat both — consistent with (C). Spot-check a rejected choice, $P = 239.95$: Coupon 3 saves $0.18 \cdot 139.95 \approx \textdollar 25.19$, which already tops Coupon 1's $\textdollar 24.00$, so (D) fails as the band predicts.
Alternative: Instead of algebra, compute all three reductions for each of the five listed prices and compare the columns. This brute-force table gives the same winner at $\textdollar 219.95$, but the inequality band $200 < P < 225$ explains *why* — and shows the answer would be unique even if the choices were different.
CCSS standards used (min grade 7)
7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (Turning each coupon's percent-off rule into an expression: $0.10P$ and $0.18(P-100)$.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Building and solving the two inequalities $0.10P > 20$ and $0.10P > 0.18(P-100)$ to get $P > 200$ and $P < 225$.)6.EE.B.5Understand solving an equation or inequality as a process of finding values that make it true (Intersecting the two conditions into $200 < P < 225$ and testing which listed price satisfies it.)
⭐ Write each coupon's savings in terms of the price, then two inequalities squeeze the price into $200 < P < 225$ — and only $\textdollar 219.95$ fits.
⭐ Write each coupon's savings in terms of the price, then two inequalities squeeze the price into $200 < P < 225$ — and only $\textdollar 219.95$ fits.
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