Fractions

What Are Fractions?

A fraction represents part of a whole. When you cut a pizza into 8 slices and eat 3 of them, you've eaten \(\frac{3}{8}\) of the pizza. The fraction tells you what portion of the total you're dealing with.

Every fraction has two parts:

• The numerator (top number) tells you how many parts you have
• The denominator (bottom number) tells you how many equal parts make up the whole

In \(\frac{3}{8}\), the numerator is 3 (you have 3 slices) and the denominator is 8 (the whole pizza was cut into 8 slices).

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

Fractions are everywhere in daily life—cooking recipes ("use \(\frac{2}{3}\) cup of flour"), measuring distances ("the screw is \(\frac{3}{16}\) of an inch"), calculating discounts ("\(\frac{1}{4}\) off the original price"), splitting bills ("you owe \(\frac{1}{3}\) of the total"), or describing progress ("we're \(\frac{3}{4}\) of the way done"). Understanding how to work with fractions isn't just about passing math class; it's a practical skill you'll use constantly.

The key is understanding that fractions represent relationships between parts and wholes. Once you grasp this concept and learn the basic operations, fractions become much less intimidating.

Types of Fractions

Not all fractions look the same. Here are the main types:

Proper fractions have a numerator smaller than the denominator: \(\frac{2}{5}\), \(\frac{7}{10}\), \(\frac{1}{3}\)

These fractions represent less than one whole. If you have \(\frac{2}{5}\) of a cake, you have less than a complete cake.

Improper fractions have a numerator greater than or equal to the denominator: \(\frac{7}{3}\), \(\frac{11}{4}\), \(\frac{5}{5}\)

These represent one whole or more. The fraction \(\frac{7}{3}\) means you have 7 thirds, which is more than one complete whole (since 3 thirds make one whole).

Mixed numbers combine a whole number with a proper fraction: \(2\frac{1}{3}\), \(5\frac{3}{4}\), \(1\frac{1}{2}\)

This notation says "2 wholes plus \(\frac{1}{3}\) of another." Mixed numbers are often easier to visualize—if someone says they ate \(2\frac{1}{2}\) pizzas, you immediately understand they ate 2 complete pizzas plus half of another.

You can convert between improper fractions and mixed numbers: $$\frac{7}{3} = 2\frac{1}{3}$$

Why? Because 7 thirds is the same as 2 wholes (which takes 6 thirds) plus 1 more third.

Equivalent Fractions

Different fractions can represent the same amount. Think about pizza again: \(\frac{1}{2}\) of a pizza is the same amount as \(\frac{2}{4}\) or \(\frac{4}{8}\). These are called equivalent fractions.

You create equivalent fractions by multiplying (or dividing) both the numerator and denominator by the same number:

Example 1: Find an equivalent fraction for \(\frac{2}{3}\).

Multiply both top and bottom by 2: $$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$$

So \(\frac{2}{3}\) and \(\frac{4}{6}\) are equivalent—they represent the same amount.

Example 2: Find an equivalent fraction for \(\frac{3}{5}\) with a denominator of 15.

We need to multiply the denominator by 3 to get 15, so multiply both parts by 3: $$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$$

Why does this work? When you multiply both parts by the same number, you're really multiplying by 1 (since \(\frac{3}{3} = 1\)), and multiplying by 1 doesn't change the value.

Simplifying Fractions

A fraction is in simplest form (or lowest terms) when the numerator and denominator have no common factors other than 1. Simplifying makes fractions easier to understand and work with.

Example 3: Simplify \(\frac{6}{8}\).

Both 6 and 8 are divisible by 2: $$\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$$

Example 4: Simplify \(\frac{12}{18}\).

Both numbers are divisible by 6: $$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$$

If you don't spot the greatest common factor right away, that's okay. You can simplify in steps: $$\frac{12}{18} = \frac{6}{9} = \frac{2}{3}$$

First divide by 2, then divide by 3. You end up at the same place.

Example 5: Simplify \(\frac{15}{40}\).

Both are divisible by 5: $$\frac{15}{40} = \frac{3}{8}$$

A quick tip: if both numbers are even, you can always divide by 2. If both numbers end in 5 or 0, you can divide by 5.

Adding and Subtracting Fractions

Adding and subtracting fractions follows one golden rule: you need a common denominator. You can't add \(\frac{1}{3}\) and \(\frac{1}{4}\) directly any more than you can add 3 inches and 4 miles—the units are different.

Same Denominator

When fractions already have the same denominator, just add or subtract the numerators:

Example 6: Add \(\frac{2}{7} + \frac{3}{7}\).

$$\frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}$$

The denominator stays the same. You're adding 2 sevenths and 3 sevenths to get 5 sevenths.

Example 7: Subtract \(\frac{5}{9} - \frac{2}{9}\).

$$\frac{5}{9} - \frac{2}{9} = \frac{5 - 2}{9} = \frac{3}{9} = \frac{1}{3}$$

Notice we simplified the answer at the end.

Different Denominators

When denominators are different, you must first rewrite the fractions with a common denominator. The easiest common denominator to find is often just the product of the two denominators.

Example 8: Add \(\frac{1}{2} + \frac{1}{3}\).

The denominators are 2 and 3. A common denominator is \(2 \times 3 = 6\).

Rewrite each fraction with denominator 6: $$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$$ $$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$$

Now add: $$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$

Example 9: Subtract \(\frac{3}{4} - \frac{1}{6}\).

Common denominator: \(4 \times 6 = 24\)

Rewrite: $$\frac{3}{4} = \frac{18}{24} \quad \text{and} \quad \frac{1}{6} = \frac{4}{24}$$

Subtract: $$\frac{18}{24} - \frac{4}{24} = \frac{14}{24} = \frac{7}{12}$$

Using the Least Common Denominator (LCD)

Sometimes using the product of the denominators gives you larger numbers than necessary. The least common denominator is the smallest number that both denominators divide into evenly.

Example 10: Add \(\frac{1}{4} + \frac{1}{6}\).

Instead of using \(4 \times 6 = 24\), notice that 12 is the smallest number divisible by both 4 and 6.

$$\frac{1}{4} = \frac{3}{12} \quad \text{and} \quad \frac{1}{6} = \frac{2}{12}$$

$$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$

Using the LCD gives you smaller numbers to work with, but using the product of denominators always works if finding the LCD feels tricky. When you're first learning, don't stress about finding the LCD—just multiply the denominators together. You can always simplify your answer at the end. As you get more comfortable, you'll start noticing common multiples more easily.

Here's a tip for finding the LCD quickly: if one denominator is a multiple of the other, the larger one is already the LCD. For example, when adding \(\frac{1}{3} + \frac{1}{6}\), notice that 6 is a multiple of 3, so 6 is your LCD.

Multiplying Fractions

Multiplying fractions is actually simpler than adding them—you don't need a common denominator! Just multiply the numerators together and multiply the denominators together.

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Example 11: Multiply \(\frac{2}{3} \times \frac{4}{5}\).

$$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$

Example 12: Multiply \(\frac{3}{7} \times \frac{2}{9}\).

$$\frac{3}{7} \times \frac{2}{9} = \frac{6}{63} = \frac{2}{21}$$

We simplified at the end by dividing both parts by 3.

Canceling Before Multiplying

You can make multiplication easier by "canceling" common factors before you multiply. This is really just simplifying early.

Example 13: Multiply \(\frac{4}{9} \times \frac{3}{8}\).

Notice that 4 and 8 share a factor of 4, and 3 and 9 share a factor of 3. Cancel these: $$\frac{4}{9} \times \frac{3}{8} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$$

This gives the same answer but with smaller numbers.

Multiplying by Whole Numbers

Remember that any whole number can be written as a fraction with denominator 1.

Example 14: Multiply \(5 \times \frac{2}{3}\).

$$5 \times \frac{2}{3} = \frac{5}{1} \times \frac{2}{3} = \frac{10}{3} = 3\frac{1}{3}$$

What does multiplication mean for fractions?

When you multiply \(\frac{1}{2} \times \frac{1}{3}\), you're finding "one-half of one-third." Imagine a candy bar divided into thirds. If you take one of those thirds and cut it in half, you get \(\frac{1}{6}\) of the original bar. That's why \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).

This is different from addition, where you combine amounts. Multiplication makes fractions smaller (usually). When you multiply \(\frac{2}{3} \times \frac{4}{5}\), you get \(\frac{8}{15}\), which is less than either starting fraction. This surprises people at first—we're used to multiplication making things bigger. But multiplying by a number less than 1 gives you less than you started with.

Dividing Fractions

Dividing fractions has a simple trick: multiply by the reciprocal. The reciprocal of a fraction is what you get when you flip it upside down.

The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
The reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).

To divide fractions: flip the second fraction and multiply.

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Example 15: Divide \(\frac{2}{3} \div \frac{4}{5}\).

Flip the second fraction and multiply: $$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$$

Example 16: Divide \(\frac{5}{8} \div \frac{1}{4}\).

$$\frac{5}{8} \div \frac{1}{4} = \frac{5}{8} \times \frac{4}{1} = \frac{20}{8} = \frac{5}{2} = 2\frac{1}{2}$$

Example 17: Divide \(3 \div \frac{2}{5}\).

Write 3 as a fraction: $$3 \div \frac{2}{5} = \frac{3}{1} \div \frac{2}{5} = \frac{3}{1} \times \frac{5}{2} = \frac{15}{2} = 7\frac{1}{2}$$

Why does this work?

Dividing by a fraction means "how many of these fit into that?" When you ask "how many \(\frac{1}{4}\)s fit into \(\frac{1}{2}\)?" the answer is 2, because two quarters make a half. The math: \(\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2\).

Here's a practical example: If you have \(\frac{3}{4}\) of a pizza and you want to divide it into servings of \(\frac{1}{8}\) pizza each, how many servings do you get? You calculate \(\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = 6\) servings. Division by fractions shows up constantly in real-world problems involving splitting things into fractional portions.

Working With Mixed Numbers

Mixed numbers like \(2\frac{1}{3}\) combine a whole number with a fraction. To add, subtract, multiply, or divide mixed numbers, it's usually easiest to convert them to improper fractions first.

Converting Mixed to Improper

Example 18: Convert \(2\frac{1}{3}\) to an improper fraction.

Multiply the whole number by the denominator, then add the numerator: $$2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}$$

Think of it this way: 2 wholes equals \(\frac{6}{3}\), plus \(\frac{1}{3}\) makes \(\frac{7}{3}\).

Example 19: Convert \(5\frac{2}{7}\) to an improper fraction.

$$5\frac{2}{7} = \frac{(5 \times 7) + 2}{7} = \frac{37}{7}$$

Converting Improper to Mixed

Example 20: Convert \(\frac{11}{4}\) to a mixed number.

Divide 11 by 4: you get 2 with a remainder of 3. $$\frac{11}{4} = 2\frac{3}{4}$$

The quotient (2) becomes the whole number, and the remainder (3) becomes the numerator.

Operations with Mixed Numbers

Example 21: Add \(2\frac{1}{3} + 1\frac{1}{2}\).

Convert to improper fractions: $$2\frac{1}{3} = \frac{7}{3} \quad \text{and} \quad 1\frac{1}{2} = \frac{3}{2}$$

Find a common denominator (6): $$\frac{7}{3} = \frac{14}{6} \quad \text{and} \quad \frac{3}{2} = \frac{9}{6}$$

Add: $$\frac{14}{6} + \frac{9}{6} = \frac{23}{6} = 3\frac{5}{6}$$

Example 22: Multiply \(2\frac{1}{2} \times 1\frac{1}{3}\).

Convert to improper fractions: $$2\frac{1}{2} = \frac{5}{2} \quad \text{and} \quad 1\frac{1}{3} = \frac{4}{3}$$

Multiply: $$\frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$$

Comparing Fractions

Which is bigger: \(\frac{3}{4}\) or \(\frac{5}{7}\)? To compare fractions, you need a common denominator.

Example 23: Which is larger: \(\frac{3}{4}\) or \(\frac{5}{7}\)?

Common denominator: 28

$$\frac{3}{4} = \frac{21}{28} \quad \text{and} \quad \frac{5}{7} = \frac{20}{28}$$

Since \(\frac{21}{28} > \frac{20}{28}\), we know \(\frac{3}{4} > \frac{5}{7}\).

Example 24: Order these from smallest to largest: \(\frac{2}{3}\), \(\frac{3}{5}\), \(\frac{5}{8}\).

Common denominator: 120

$$\frac{2}{3} = \frac{80}{120}, \quad \frac{3}{5} = \frac{72}{120}, \quad \frac{5}{8} = \frac{75}{120}$$

Ordered: \(\frac{3}{5} < \frac{5}{8} < \frac{2}{3}\)

Cross-multiplication shortcut for comparing two fractions: multiply diagonally and compare.

For \(\frac{3}{4}\) vs \(\frac{5}{7}\): \(3 \times 7 = 21\) and \(4 \times 5 = 20\). Since \(21 > 20\), we know \(\frac{3}{4} > \frac{5}{7}\).

Practical Tips and Shortcuts

After working through all these operations, here are some strategies to make fractions easier:

Always simplify your final answer. Even if the problem doesn't ask for it, get in the habit of writing fractions in simplest form. It makes your answers cleaner and easier to check.

Check if you can cancel before multiplying. This saves you from dealing with large numbers and having to simplify at the end. If you see \(\frac{6}{7} \times \frac{14}{15}\), notice that 6 and 15 share a factor of 3, and 7 and 14 share a factor of 7. Cancel first to get \(\frac{2}{1} \times \frac{2}{5} = \frac{4}{5}\).

For division, remember: "flip and multiply." This rhyme has helped countless students remember the rule for dividing fractions.

When adding or subtracting, don't add the denominators. This is one of the most common mistakes. If you have \(\frac{1}{2} + \frac{1}{3}\), the answer is NOT \(\frac{2}{5}\). You need that common denominator first.

Convert mixed numbers for multiplication and division. While you can add and subtract mixed numbers directly (by handling whole numbers and fractions separately), it's usually easier to convert to improper fractions for multiplication and division.

Use estimation to check if your answer makes sense. If you're adding \(\frac{1}{2} + \frac{1}{3}\) and you get \(\frac{5}{6}\), that makes sense—it's close to 1, which is roughly what \(\frac{1}{2} + \frac{1}{3}\) should be. If you got \(\frac{2}{5}\), that's less than \(\frac{1}{2}\) alone, so something went wrong.

Try These Problems

Work through these, then check your answers below.

1. Simplify \(\frac{24}{36}\)

2. Add \(\frac{2}{5} + \frac{1}{3}\)

3. Subtract \(\frac{7}{8} - \frac{1}{2}\)

4. Multiply \(\frac{3}{4} \times \frac{2}{9}\)

5. Divide \(\frac{2}{3} \div \frac{4}{5}\)

6. Convert \(3\frac{2}{5}\) to an improper fraction

7. Convert \(\frac{17}{4}\) to a mixed number

8. Add \(1\frac{1}{2} + 2\frac{1}{4}\)

9. Multiply \(3 \times \frac{2}{7}\)

10. Which is larger: \(\frac{4}{5}\) or \(\frac{7}{9}\)?

Check Your Answers

1. \(\frac{2}{3}\)
   Divide both by 12: \(\frac{24 \div 12}{36 \div 12} = \frac{2}{3}\)

2. \(\frac{11}{15}\)
   Common denominator 15: \(\frac{6}{15} + \frac{5}{15} = \frac{11}{15}\)

3. \(\frac{3}{8}\)
   Common denominator 8: \(\frac{7}{8} - \frac{4}{8} = \frac{3}{8}\)

4. \(\frac{1}{6}\)
   \(\frac{3 \times 2}{4 \times 9} = \frac{6}{36} = \frac{1}{6}\)

5. \(\frac{5}{6}\)
   \(\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\)

6. \(\frac{17}{5}\)
   \((3 \times 5) + 2 = 17\), so \(\frac{17}{5}\)

7. \(4\frac{1}{4}\)
   \(17 \div 4 = 4\) remainder \(1\)

8. \(3\frac{3}{4}\)
   \(\frac{3}{2} + \frac{9}{4} = \frac{6}{4} + \frac{9}{4} = \frac{15}{4} = 3\frac{3}{4}\)

9. \(\frac{6}{7}\)
   \(\frac{3}{1} \times \frac{2}{7} = \frac{6}{7}\)

10. \(\frac{7}{9}\)
    \(\frac{4}{5} = \frac{36}{45}\) and \(\frac{7}{9} = \frac{35}{45}\)... wait, let me recalculate: \(\frac{4}{5} = 0.8\) and \(\frac{7}{9} \approx 0.778\), so \(\frac{4}{5}\) is larger

Why Fractions Matter

Fractions are fundamental to mathematics and everyday life. They show up in cooking (half a cup, three-quarters of a teaspoon), construction (two and a quarter inches), sports (a quarterback's completion percentage), finance (interest rates, stock prices), and countless other situations.

More importantly, fractions are the foundation for algebra. When you work with rational expressions, you're using all these same fraction skills—just with variables instead of numbers. Understanding fractions deeply now makes algebra much easier later.

The key to mastering fractions is practice. Work through problems until the procedures become automatic. Once you can add, subtract, multiply, and divide fractions without thinking hard about the steps, you'll find that many more advanced math topics become much more accessible.