2004 AMC 10A Problem 3

P1-10~1 minPrint

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

Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?

Show solution without answering

Key insight

The answer must be in cents, so change 20 dollars into 2000 cents first. Then 1.45 percent of 2000 is 29.

Solution

1. The question asks for cents, so switch to cents right away. One dollar is $100$ cents, so $20$ dollars is $2000$ cents.
2. Write the percent as a decimal: $1.45\% = \dfrac{1.45}{100} = 0.0145$ .
3. Multiply:
$0.0145 \times 2000 = 29.$

So $29$ cents of each hour's pay goes to local taxes.

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

Why this works

Taking a percent of something is multiplying by a decimal. The only real decision is which unit to use, and using the unit the question asks for from the start avoids an error of $100$ at the end. When the answer choices differ only by moving the decimal point, the problem is testing units.

Alternative approach

Estimate. $1\%$ of $2000$ cents is $20$ cents, so $1.45\%$ should be a little more than $20$ cents. Only $29$ , choice (E), is anywhere close.

Concise solution

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

The question asks for cents, so express the wage in cents before taking the percentage: $20$ dollars is $2000$ cents.

Now $1.45\% = 0.0145$ , and
$0.0145 \times 2000 = 29.$

The tax withheld is $29$ cents per hour, so the answer is $\boxed{\textbf{(E)}\ 29}$ .

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)