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 $n?$

$\ \ 27$

$\ 28$

$\ 29$

$\ 31$

$\ 33$

Answer: C

Concepts:percentage inequality

Difficulty rating: 1540

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 .

2014 AMC 10B Problem 11

Answer: C

Difficulty rating: 1540

Solution:

Problem 11 in Other Years