Fractions, Decimals, and Percents

Fractions, decimals, and percents are three different ways to represent the same thing: parts of a whole. You might see $\frac{1}{2}$, 0.5, and 50% used interchangeably because they all mean exactly the same amount. Being able to move fluently between these three forms is essential in algebra, and it's a skill that will serve you well in everyday life too.

Think of it like speaking three different languages that all describe the same concepts. Sometimes fractions are more convenient (like when you're working with recipes), sometimes decimals make more sense (like when you're measuring), and sometimes percents are clearer (like when you're talking about grades or discounts). The math doesn't care which form you use, but often one form makes a problem easier to solve than the others.

Understanding Fractions

A fraction represents a part of a whole. The number on top is the numerator (how many parts you have), and the number on bottom is the denominator (how many parts make up the whole).

$$\frac{\text{numerator}}{\text{denominator}}$$

Example: $\frac{3}{4}$ means you have 3 parts out of 4 total parts. If you cut a pizza into 4 slices and eat 3 of them, you've eaten $\frac{3}{4}$ of the pizza.

The denominator tells you how many equal pieces something has been divided into. The numerator tells you how many of those pieces you're talking about. So $\frac{5}{8}$ means something was divided into 8 equal parts, and you have 5 of them.

Here's another way to think about it: the fraction bar means division. So $\frac{3}{4}$ is really just another way to write $3 \div 4$. This is why when you convert a fraction to a decimal, you divide the top by the bottom.

Fractions can be proper (numerator smaller than denominator, like $\frac{2}{5}$) or improper (numerator larger than or equal to denominator, like $\frac{7}{3}$). A proper fraction represents less than one whole. An improper fraction represents one whole or more, and can be converted to a mixed number: $\frac{7}{3} = 2\frac{1}{3}$ (you have 2 whole things plus $\frac{1}{3}$ of another).

Understanding Decimals

Decimals are another way to represent parts of a whole, using place value. The decimal point separates the whole number part from the fractional part.

$$3.75 = 3 + \frac{7}{10} + \frac{5}{100}$$

The digits after the decimal point represent tenths, hundredths, thousandths, and so on. Each place to the right of the decimal point is worth one-tenth of the place before it. The first place after the decimal is tenths, the second is hundredths, the third is thousandths.

Think about money: $3.75 means 3 dollars and 75 cents. Since there are 100 cents in a dollar, those 75 cents are $\frac{75}{100}$ of a dollar. The decimal system works the same way.

Example: 0.6 means 6 tenths, or $\frac{6}{10}$. If you have a dollar and spend 60 cents, you spent 0.6 of your dollar.

Example: 0.25 means 25 hundredths, or $\frac{25}{100}$. This is the same as one quarter (25 cents out of 100 cents).

When you see a number like 2.8, read it as "two and eight tenths." The whole number part (2) comes before the decimal point, and the fractional part (8 tenths) comes after it.

Understanding Percents

Percent means "per hundred" or "out of 100." When you see 75%, it means 75 out of 100, or $\frac{75}{100}$.

The word "percent" literally breaks down as "per cent" where "cent" means 100 (like a century is 100 years, or 100 cents make a dollar). So when you see the % symbol, think "out of 100."

The percent symbol (%) is just shorthand for "divide by 100." So 75% is really just another way to write 0.75 or $\frac{3}{4}$. All three of these mean exactly the same amount.

Example: If you scored 85% on a test, you got 85 points out of a possible 100 points. If the test had 20 questions, you got 17 of them correct (since $\frac{17}{20} = \frac{85}{100}$).

Percents are useful because they give us a common way to compare things. If one test was out of 50 points and another was out of 80 points, comparing your raw scores wouldn't be fair. But if you got 80% on both tests, you know you did equally well.

You can have percents greater than 100%. If something is 150%, that means it's one and a half times the original amount. If a population grows by 200%, it's now three times as large as it started (the original 100% plus an additional 200%).

Converting Between Forms

Being able to convert between fractions, decimals, and percents is crucial. Let's look at each conversion.

Fraction to Decimal

To convert a fraction to a decimal, divide the numerator by the denominator.

Example: Convert $\frac{3}{4}$ to a decimal.

$$\frac{3}{4} = 3 \div 4 = 0.75$$

Example: Convert $\frac{1}{8}$ to a decimal.

$$\frac{1}{8} = 1 \div 8 = 0.125$$

Some fractions create repeating decimals. For instance, $\frac{1}{3} = 0.333...$ where the 3 repeats forever. We often write this as $0.\overline{3}$ with a bar over the repeating digit.

Decimal to Fraction

To convert a decimal to a fraction, write it as a fraction with a denominator that's a power of 10, then simplify.

Example: Convert 0.6 to a fraction.

0.6 means 6 tenths: $$0.6 = \frac{6}{10} = \frac{3}{5}$$

Example: Convert 0.75 to a fraction.

0.75 means 75 hundredths: $$0.75 = \frac{75}{100} = \frac{3}{4}$$

Example: Convert 0.125 to a fraction.

0.125 means 125 thousandths: $$0.125 = \frac{125}{1000} = \frac{1}{8}$$

Percent to Decimal

To convert a percent to a decimal, divide by 100 (or just move the decimal point two places to the left).

Example: Convert 45% to a decimal.

$$45\% = \frac{45}{100} = 0.45$$

Example: Convert 7% to a decimal.

$$7\% = \frac{7}{100} = 0.07$$

Example: Convert 150% to a decimal.

$$150\% = \frac{150}{100} = 1.50 = 1.5$$

Decimal to Percent

To convert a decimal to a percent, multiply by 100 (or just move the decimal point two places to the right) and add the % symbol.

Example: Convert 0.38 to a percent.

$$0.38 = 0.38 \times 100 = 38\%$$

Example: Convert 0.9 to a percent.

$$0.9 = 0.9 \times 100 = 90\%$$

Example: Convert 1.25 to a percent.

$$1.25 = 1.25 \times 100 = 125\%$$

Fraction to Percent

To convert a fraction to a percent, first convert the fraction to a decimal, then convert the decimal to a percent.

Example: Convert $\frac{3}{5}$ to a percent.

Step 1: Convert to decimal: $\frac{3}{5} = 0.6$

Step 2: Convert to percent: $0.6 = 60\%$

Example: Convert $\frac{7}{8}$ to a percent.

Step 1: $\frac{7}{8} = 0.875$

Step 2: $0.875 = 87.5\%$

Percent to Fraction

To convert a percent to a fraction, write it as a fraction over 100 and simplify.

Example: Convert 40% to a fraction.

$$40\% = \frac{40}{100} = \frac{2}{5}$$

Example: Convert 75% to a fraction.

$$75\% = \frac{75}{100} = \frac{3}{4}$$

Working With Fractions

When adding or subtracting fractions, you need a common denominator.

Example: Add $\frac{1}{4} + \frac{1}{2}$

First, find a common denominator (4 works): $$\frac{1}{4} + \frac{2}{4} = \frac{3}{4}$$

Example: Subtract $\frac{5}{6} - \frac{1}{3}$

Common denominator is 6: $$\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$$

When multiplying fractions, multiply the numerators together and the denominators together.

Example: Multiply $\frac{2}{3} \times \frac{3}{4}$

$$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$$

When dividing fractions, multiply by the reciprocal (flip the second fraction).

Example: Divide $\frac{2}{3} \div \frac{4}{5}$

$$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$$

Working With Decimals

Adding and subtracting decimals works just like whole numbers—just line up the decimal points.

Example: $3.7 + 2.48$

$$\begin{align} &3.70 \

&2.48 \ \hline &6.18 \end{align}$$

For multiplication, multiply as if there were no decimal points, then count the total number of decimal places in both numbers and place the decimal point in your answer.

Example: $2.5 \times 1.2$

$$25 \times 12 = 300$$

There are 2 decimal places total (one in 2.5, one in 1.2), so: $$2.5 \times 1.2 = 3.00 = 3.0$$

Working With Percents

To find a percent of a number, convert the percent to a decimal and multiply.

Example: What is 30% of 80?

$$30\% = 0.30$$ $$0.30 \times 80 = 24$$

Example: What is 15% of 60?

$$15\% = 0.15$$ $$0.15 \times 60 = 9$$

To find what percent one number is of another, divide the part by the whole and convert to a percent.

Example: 12 is what percent of 48?

$$\frac{12}{48} = 0.25 = 25\%$$

Common Equivalents to Memorize

Some conversions come up so often that it's worth memorizing them:

$\frac{1}{2} = 0.5 = 50\%$
$\frac{1}{4} = 0.25 = 25\%$
$\frac{3}{4} = 0.75 = 75\%$
$\frac{1}{3} = 0.\overline{3} = 33.\overline{3}\%$
$\frac{2}{3} = 0.\overline{6} = 66.\overline{6}\%$
$\frac{1}{5} = 0.2 = 20\%$
$\frac{1}{10} = 0.1 = 10\%$
$\frac{1}{8} = 0.125 = 12.5\%$

Knowing these by heart will speed up your work considerably.

Practice Problems

Try these conversions and calculations:

Convert $\frac{3}{8}$ to a decimal
Convert 0.35 to a fraction (simplified)
Convert 68% to a decimal
Convert 0.8 to a percent
What is 25% of 80?
15 is what percent of 60?
Add: $\frac{2}{5} + \frac{1}{4}$
Multiply: $\frac{3}{7} \times \frac{2}{3}$

Solutions

$\frac{3}{8} = 3 \div 8 = 0.375$
$0.35 = \frac{35}{100} = \frac{7}{20}$
$68\% = 0.68$
$0.8 = 80\%$
$0.25 \times 80 = 20$
$\frac{15}{60} = 0.25 = 25\%$
$\frac{2}{5} + \frac{1}{4} = \frac{8}{20} + \frac{5}{20} = \frac{13}{20}$
$\frac{3}{7} \times \frac{2}{3} = \frac{6}{21} = \frac{2}{7}$

What's Next?

You've completed the foundational unit! You now understand variables and expressions, the order of operations, properties of numbers, different types of numbers, absolute value, and how to work with fractions, decimals, and percents. These are the building blocks for everything else in algebra.

In the next unit, we'll start working with algebraic expressions—simplifying them, combining like terms, and using the distributive property to manipulate equations. This is where algebra really starts to come alive.