2014 AMC 10B Problem 11

Question

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)

A

25\%

discount followed by a

5\%

discount.

What is the smallest possible positive integer value of

n?

\ \ 27

\ 28

\ 29

\ 31

\ 33

Convert each stacked discount into the final fraction of the price paid

The single discount must exceed the largest of the three effective discounts

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 .

Problem 11 in other years