2014 AMC 10B Problem 11

P11-15~3 minPrint

Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

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 $n$ ?

Show solution without answering

Key insight

Discounts one after another multiply what is left of the price. The three deals take off 27.75%, 27.1% and 28.75%, so n must be more than 28.75.

Solution

Use a price of $\$100$ , so every percent is just a number of dollars.

Two $15\%$ discounts. $15\%$ off $\$100$ leaves $\$85$ . Another $15\%$ off $\$85$ takes off $\$12.75$ , leaving $\$72.25$ . Total discount: $\$27.75$ , or $27.75\%$ .
Three $10\%$ discounts. $\$100 \to \$90 \to \$81 \to \$72.90$ . Total discount: $27.1\%$ .
$25\%$ then $5\%$ . $\$100 \to \$75$ , then $5\%$ of $\$75$ is $\$3.75$ , leaving $\$71.25$ . Total discount: $28.75\%$ .
The single discount of $n\%$ must be more advantageous than every one of these, so it must take off more than the biggest of them: $n > 28.75$ .
The smallest whole number bigger than $28.75$ is $29$ .

The answer is $\boxed{\textbf{(C)}\ 29}$ .

Why this works

The second discount is taken from a price that is already lower, so two $15\%$ discounts take off less than $30\%$ . To compare deals, track how much of the price is left after each one; multiplying the "left over" parts ( $0.85 \times 0.85$ , and so on) does this quickly. Also read the inequality carefully: "more advantageous than" is strict, so $n$ has to beat $28.75$ , not tie it.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Compute the fraction of the price that remains after each scheme.

(1) $0.85 \cdot 0.85 = 0.7225$ , an effective discount of $27.75\%$ .

(2) $0.9^3 = 0.729$ , an effective discount of $27.1\%$ .

(3) $0.75 \cdot 0.95 = 0.7125$ , an effective discount of $28.75\%$ .

The single discount must be strictly better than all three, so $n > 28.75$ . The smallest integer is $n = 29$ .

The answer is $\boxed{\textbf{(C)}\ 29}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)