2009 AMC 10B Problem 5

P1-10~1 minPrint

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

Twenty percent less than 60 is one-third more than what number?

Show solution without answering

Key insight

Twenty percent less than 60 is 48. A number plus a third of itself is 4 thirds of it, so 4 thirds make 48.

Solution

Work out the left side first, since it is just a number.

Twenty percent of $60$ is $12$ . Twenty percent less than $60$ is $60 - 12 = 48$ .
Now we need a number that becomes $48$ when one-third of it is added on.
Think of the unknown number as $3$ equal parts. One-third of it is $1$ part. So "one-third more" is $3 + 1 = 4$ parts.
Those $4$ parts make $48$ , so each part is $12$ , and the number (3 parts) is $36$ .
Check: one-third of $36$ is $12$ , and $36 + 12 = 48$ . Correct.

The answer is $\boxed{\textbf{(D)}\ 36}$ .

Why this works

"Twenty percent less" and "one-third more" both change a number by a fraction of itself. Here the one-third is a third of the unknown number, not of $48$ . Thinking in equal parts ( $3$ parts grow to $4$ parts) keeps that straight and avoids the tempting wrong answer $32$ .

Concise solution

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

Twenty percent less than $60$ is $60 \cdot 0.8 = 48$ .

If the unknown number is $n$ , then one-third more than $n$ is $n + \frac{n}{3} = \frac{4n}{3}$ . Setting $\frac{4n}{3} = 48$ gives $n = 48 \cdot \frac34 = 36$ .

Check: one-third of $36$ is $12$ , and $36 + 12 = 48$ .

The answer is $\boxed{\textbf{(D)}\ 36}$ .

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)