Lesson 2.2: More Fractions and Percents

New Concept

A percent is a fraction "out of 100." This lesson shows how to convert between percents, decimals, and common fractions like eighths and thirds. Mastering these conversions will help you easily calculate tips, sales tax, and discounts.

What’s next

Next, you'll tackle worked examples and practice cards to master percent calculations. Get ready to apply these skills in interactive budgeting problems.

Property

"Percent" means "for each 100" or "out of 100." So a percent is just a fraction whose denominator is 100. For example, $75\% = 0.75 = \frac{75}{100}$. Because 10% is equal to $\frac{1}{10}$, it is easy to find 10% of a number: we just divide the number by 10.

Examples

• A store offers a 10% discount on a 90 dollar jacket. The discount is $\frac{1}{10}$ of 90 dollars, which is $90 \div 10 = 9$ dollars.
• To find 20% of 500, first find 10% by dividing by 10, which is $500 \div 10 = 50$. Then, double it for 20%, so $2 \times 50 = 100$.
• A restaurant bill is 40 dollars. A 15% tip can be found by taking 10% (4 dollars) and adding 5% (half of 10%, which is 2 dollars). The total tip is $4 + 2 = 6$ dollars.

Explanation

Think of the percent sign (%) as shorthand for "out of 100." It's a simple way to discuss parts of a whole. Knowing shortcuts, like 10% being one-tenth of a number, makes mental math much faster and easier.

Property

To calculate a percent of a number:

1. Change the percent to a decimal fraction.
2. Multiply the number by the decimal fraction.

To change a percent to a decimal fraction, divide the percent by 100, or move the decimal point two places to the left.

Examples

• To find 8% of 300, convert 8% to a decimal by moving the decimal point two places left: $8\% = 0.08$. Then multiply: $0.08 \times 300 = 24$.
• A phone costs 850 dollars and sales tax is 7.5%. Convert 7.5% to a decimal: $0.075$. The tax is $0.075 \times 850 = 63.75$ dollars.
• Calculate 150% of 60. First, change 150% to a decimal: $1.50$. Then, multiply: $1.50 \times 60 = 90$.

Explanation

To use a percent in a calculation, you must first convert it into a form your calculator understands, like a decimal. Moving the decimal point two places left is a quick shortcut for dividing by 100.

Property

To find what percent a part is of a whole:

1. Divide the part by the whole to get a decimal fraction:
   $$\frac{\text{part}}{\text{whole}}$$
2. Multiply the decimal fraction by 100 to convert it to a percent.

Examples

• You scored 18 points out of a total of 20 on a quiz. To find your percent score, divide the part by the whole: $18 \div 20 = 0.90$. Multiply by 100 to get the percent: $0.90 \times 100 = 90\%$.
• A recipe calls for 400 grams of flour, and you have used 80 grams. The percent used is $80 \div 400 = 0.20$. This is equivalent to $20\%$.
• If you saved 30 dollars on an item that originally cost 150 dollars, you saved $\frac{30}{150} = 0.2$. As a percent, this is $0.2 \times 100 = 20\%$.

Explanation

This process reverses the calculation. By dividing the smaller part by the total whole, you find the fraction it represents. Multiplying by 100 scales that fraction up to a percentage, making it easier to understand.

Property

The fraction $\frac{1}{8}$ is one-half of $\frac{1}{4}$. Since $\frac{1}{4} = 0.25$, the decimal form of $\frac{1}{8}$ is $0.125$. This is equivalent to $12.5\%$. Other eighths are multiples of this value. For example, $\frac{3}{8} = 3 \times 0.125 = 0.375$, which is $37.5\%$.

Examples

• To calculate 62.5% of 400, recognize that $62.5\%$ is $\frac{5}{8}$. First, find $\frac{1}{8}$ of 400 by dividing: $400 \div 8 = 50$. Then multiply by 5: $5 \times 50 = 250$.
• A bolt is $\frac{7}{8}$ of an inch. To express this as a decimal, multiply $7 \times 0.125$, since $\frac{1}{8} = 0.125$. The result is $0.875$ inches.
• A company wants to allocate 37.5% of its 8000 dollar budget to marketing. Since $37.5\% = \frac{3}{8}$, the amount is $\frac{3}{8} \times 8000 = 3 \times (8000 \div 8) = 3 \times 1000 = 3000$ dollars.

Explanation

Knowing the decimal and percent for $\frac{1}{8}$ is a powerful shortcut. You can find any multiple of one-eighth, like $\frac{3}{8}$ or $\frac{5}{8}$, by simple multiplication, often without a calculator. This helps in budgeting and measurements.

Property

The fraction $\frac{1}{3}$ has a repeating decimal form: $0.333...$ or $0.\overline{3}$. We often approximate it as $0.33$. The exact percent form is $33\frac{1}{3}\%$. For $\frac{2}{3}$, the decimal is $0.666...$ or $0.\overline{6}$, and the exact percent is $66\frac{2}{3}\%$.

Examples

• A baker needs to find $\frac{1}{3}$ of a 150-pound bag of flour. This is $150 \div 3 = 50$ pounds. This is exactly $33\frac{1}{3}\%$ of the bag.
• To find $66\frac{2}{3}\%$ of 90 dollars, recognize this percent as the fraction $\frac{2}{3}$. The amount is $\frac{2}{3} \times 90 = 2 \times (90 \div 3) = 2 \times 30 = 60$ dollars.
• A project is estimated to take 12 hours. If you have completed $\frac{1}{3}$ of the work, you have worked $12 \div 3 = 4$ hours. The decimal representation is approximately $0.33$ of the total time.

Explanation

Not all fractions have clean decimal endings. Thirds are a classic example of repeating decimals. Using mixed-number percents like $33\frac{1}{3}\%$ allows for perfect accuracy without rounding, which is important in many calculations.