1.6 Add and Subtract Fractions

Adding and Subtracting Fractions

Adding and subtracting fractions is one of the core skills you'll use throughout algebra. Whether you're simplifying expressions or solving equations, fractions show up constantly. This section covers how to handle common and different denominators, simplify complex fractions, and solve equations that contain fractions.

Adding Fractions with Common Denominators

When two fractions already share the same denominator, the process is straightforward: add the numerators and keep the denominator the same. Then simplify if possible.

$\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}$

To simplify, divide the numerator and denominator by their greatest common factor (GCF).

• $\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$ (GCF of 4 and 8 is 4)
• $\frac{5}{12} + \frac{7}{12} = \frac{12}{12} = 1$

Subtraction works the same way: subtract the numerators and keep the denominator.

Adding Fractions with Different Denominators

When the denominators don't match, you need to rewrite each fraction so they share a common denominator before you can add or subtract.

1. Find the least common multiple (LCM) of the two denominators. This becomes your common denominator.
2. Multiply the numerator and denominator of each fraction by whatever factor makes its denominator equal to the LCM.
3. Add (or subtract) the numerators. Keep the common denominator.
4. Simplify the result by dividing numerator and denominator by their GCF.

Example 1: $\frac{1}{4} + \frac{1}{6}$

The LCM of 4 and 6 is 12. Multiply $\frac{1}{4}$ by $\frac{3}{3}$ and $\frac{1}{6}$ by $\frac{2}{2}$:

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

Example 2: $\frac{2}{3} + \frac{3}{5}$

The LCM of 3 and 5 is 15:

$\frac{10}{15} + \frac{9}{15} = \frac{19}{15}$

This result is an improper fraction (the numerator is larger than the denominator), which is perfectly valid. You could also write it as the mixed number $1\frac{4}{15}$.

Simplifying Complex Fractions

A complex fraction has a fraction in its numerator, its denominator, or both. To simplify one:

1. Simplify the numerator into a single fraction.
2. Simplify the denominator into a single fraction.
3. Divide the numerator fraction by the denominator fraction (multiply by the reciprocal).
4. Simplify the result.

Example 1:

$\frac{\frac{1}{2} + \frac{1}{3}}{\frac{2}{5} - \frac{1}{6}}$

• Numerator: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
• Denominator: $\frac{2}{5} - \frac{1}{6} = \frac{12}{30} - \frac{5}{30} = \frac{7}{30}$
• Divide: $\frac{5}{6} \div \frac{7}{30} = \frac{5}{6} \cdot \frac{30}{7} = \frac{150}{42} = \frac{25}{7}$

Example 2:

$\frac{\frac{3}{4} - \frac{1}{6}}{\frac{2}{3} + \frac{1}{2}}$

• Numerator: $\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$
• Denominator: $\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$
• Divide: $\frac{7}{12} \div \frac{7}{6} = \frac{7}{12} \cdot \frac{6}{7} = \frac{42}{84} = \frac{1}{2}$

Solving Equations with Fractions

When an equation contains fractions, you can clear them out by multiplying every term on both sides by the least common denominator (LCD). This turns the equation into one with whole numbers, which is much easier to solve.

1. Identify the LCD of all the fractions in the equation.
2. Multiply every term on both sides by the LCD.
3. Solve the resulting equation using standard algebra (isolate the variable).

Example 1: $\frac{2x}{3} + \frac{1}{4} = \frac{5}{6}$

The LCD of 3, 4, and 6 is 12. Multiply every term by 12:

• $12 \cdot \frac{2x}{3} + 12 \cdot \frac{1}{4} = 12 \cdot \frac{5}{6}$
• $8x + 3 = 10$
• Subtract 3: $8x = 7$
• Divide by 8: $x = \frac{7}{8}$

Example 2: $\frac{3}{4}x - \frac{1}{2} = \frac{1}{3}$

The LCD of 4, 2, and 3 is 12:

• $12 \cdot \frac{3}{4}x - 12 \cdot \frac{1}{2} = 12 \cdot \frac{1}{3}$
• $9x - 6 = 4$
• Add 6: $9x = 10$
• Divide by 9: $x = \frac{10}{9}$

Working with Mixed Numbers and Equivalent Fractions

Mixed numbers combine a whole number with a proper fraction, like $3\frac{1}{2}$. To add or subtract mixed numbers, convert them to improper fractions first. For $3\frac{1}{2}$, multiply the whole number by the denominator and add the numerator: $\frac{(3 \times 2) + 1}{2} = \frac{7}{2}$. Now you can use the standard addition and subtraction methods.

Equivalent fractions represent the same value even though they look different. For example, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent because $1 \times 4 = 2 \times 2$. You can test this with cross-multiplication: if $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent, then $a \times d = b \times c$.

Reciprocals are what you get when you flip a fraction's numerator and denominator. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. Reciprocals matter because dividing by a fraction is the same as multiplying by its reciprocal. That's exactly the technique you used above when simplifying complex fractions.