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2014 AMC 10A Problem 11

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2014 AMC 10A Problem 11AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 3 min · position P11-15

A 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 $\$50$

Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$

Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$

For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$ ?

A)
$179.95
B)
$199.95
C)
$219.95
D)
$239.95
E)
$259.95

Answer

Key insight

Coupon 1 beats coupon 2 when 0.1P > 20 (P > 200) and beats coupon 3 when 0.1P > 0.18(P - 100) (P < 225); only 219.95 fits.

Solution

Let the listed price be $P$ ; every choice exceeds $\$100$ , so all three coupons apply. The savings are
$\text{Coupon 1: } 0.1P,\qquad \text{Coupon 2: } 20,\qquad \text{Coupon 3: } 0.18(P - 100).$

Coupon 1 must beat coupon 2:
$0.1P > 20 \quad\Longrightarrow\quad P > 200.$

Coupon 1 must beat coupon 3:
$0.1P > 0.18P - 18 \quad\Longrightarrow\quad 18 > 0.08P \quad\Longrightarrow\quad P < 225.$

So the price must satisfy $200 < P < 225$ . Among the choices only $\$219.95$ lies in that range.

The answer is $\boxed{\textbf{(C)}\ \$219.95}$ .

Why this works

A percentage discount grows with the price while a flat discount does not, and a discount on the excess over $\$100$ grows faster than $10\%$ once the price is large enough. Translating each comparison into a linear inequality yields a window for $P$ ; the lower bound comes from the flat coupon and the upper bound from the steeper coupon.

Alternative approach

Test the choices directly. At $\$219.95$ : coupon 1 saves $\$22.00$ , coupon 2 saves $\$20$ , coupon 3 saves $0.18 \cdot 119.95 \approx \$21.59$ . At $\$239.95$ coupon 3 saves about $\$25.19$ , beating coupon 1's $\$24.00$ ; at $\$199.95$ coupon 1 saves under $\$20$ .

The trap

Comparing the final prices incorrectly, or only checking coupon 1 against coupon 2 and picking the first choice above 200.

Common mistakes

Comparing the final prices incorrectly, or only checking coupon 1 against coupon 2 and picking the first choice above 200.
Computing coupon 3 as $18\%$ of the whole price instead of $18\%$ of the amount over $\$100$ .

Techniques

Use the answer choices (mod checks, size, form) to eliminate or select · Set up the equation/formula and compute; no special trick needed