2014 AMC 10B Problem 11

Problem 11 of 25IntermediateAlgebra

For the consumer, a single discount of

n\%

is more advantageous than any of the following discounts:

(1)

Two successive

15\%

discounts.

(2)

Three successive

10\%

discounts.

(3)

25\%

discount followed by a

5\%

discount. What is the smallest possible positive integer value of

n?

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Solution

The three successive-discount plans leave the customer paying

\begin{aligned} 0.85^2&=0.7225,\\ 0.9^3&=0.729,\\ 0.75(0.95)&=0.7125 \end{aligned}

times the listed price. Their effective discounts are therefore

27.75\%

27.1\%

, and

28.75\%

. A single

n\%

discount must exceed all three, so

n>28.75

. The smallest positive integer that works is

29

. Thus, the correct answer is C .