Lesson 1.6: Add and Subtract Fractions

New Concept

Mastering fraction addition and subtraction is essential. This lesson shows how to combine fractions, whether they share a denominator or require finding a least common denominator (LCD). You'll apply this to simplify complex fractions and evaluate expressions.

What’s next

Next, you’ll work through interactive examples of adding and subtracting fractions, then apply your skills in a series of practice problems and video breakdowns.

Property

If $a$, $b$, and $c$ are numbers where $c \neq 0$, then

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

Examples

• Find the sum $\frac{y}{5} + \frac{4}{5}$. Since the denominators are the same, we add the numerators: $\frac{y+4}{5}$.
• Find the difference $-\frac{21}{25} - \frac{14}{25}$. We subtract the numerators: $\frac{-21-14}{25} = \frac{-35}{25}$, which simplifies to $-\frac{7}{5}$.
• Simplify: $\frac{15}{z} - \frac{4}{z}$. With a common denominator $z$, we get $\frac{15-4}{z} = \frac{11}{z}$.

Explanation

When fractions have the same denominator, their pieces are the same size. This means you can simply combine the numerators (the top numbers) and keep the denominator (the bottom number) the same. It's like adding 2 apples and 3 apples.

Property

To add or subtract fractions with different denominators, first find the least common denominator (LCD). The LCD is the least common multiple (LCM) of the denominators. Then, convert each fraction into an equivalent fraction with the LCD. Finally, add or subtract the numerators and place the result over the common denominator. Do not simplify the equivalent fractions before combining them, or you will lose the common denominator.

Examples

• To add $\frac{3}{10} + \frac{7}{15}$, the LCD of 10 and 15 is 30. We rewrite the fractions as $\frac{9}{30} + \frac{14}{30} = \frac{23}{30}$.
• To subtract $\frac{5}{12} - \frac{9}{16}$, the LCD is 48. This becomes $\frac{20}{48} - \frac{27}{48} = \frac{-7}{48}$.
• To add $\frac{4}{9} + \frac{x}{2}$, the LCD is 18. We get $\frac{8}{18} + \frac{9x}{18} = \frac{8+9x}{18}$.

Explanation

You can't add halves and thirds directly. You must first find a common size for the pieces. Finding the least common denominator (LCD) lets you rewrite the fractions so they can be combined, like turning different coins into pennies before counting.

Property

To simplify a complex fraction (a fraction containing other fractions):

1. Simplify the numerator into a single fraction.
2. Simplify the denominator into a single fraction.
3. Divide the numerator by the denominator. Remember that dividing by a fraction is the same as multiplying by its reciprocal. Simplify the result if possible.

Examples

• Simplify $\frac{(\frac{1}{3})^2}{5+2^2}$. The numerator is $\frac{1}{9}$ and the denominator is $9$. So, $\frac{\frac{1}{9}}{9} = \frac{1}{9} \cdot \frac{1}{9} = \frac{1}{81}$.
• Simplify $\frac{\frac{1}{2}+\frac{1}{4}}{\frac{2}{3}-\frac{1}{6}}$. The numerator simplifies to $\frac{3}{4}$ and the denominator to $\frac{3}{6} = \frac{1}{2}$. The problem becomes $\frac{\frac{3}{4}}{\frac{1}{2}} = \frac{3}{4} \cdot \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$.
• Simplify $\frac{5}{\frac{1}{2}+\frac{1}{3}}$. The denominator is $\frac{3}{6}+\frac{2}{6} = \frac{5}{6}$. So we have $\frac{5}{\frac{5}{6}} = 5 \cdot \frac{6}{5} = 6$.

Explanation

A complex fraction is just a division problem in disguise. The main fraction bar means 'divide'. Clean up the top expression and the bottom expression first, turning each into a single fraction. Then, solve by dividing the top by the bottom.

Property

To evaluate a variable expression with fractions, substitute the given fraction for each variable in the expression. Then, use the order of operations (PEMDAS/BODMAS) to simplify the resulting numerical expression. This often involves adding, subtracting, multiplying, or dividing the fractions as needed.

Examples

• Evaluate $x + \frac{3}{5}$ when $x = -\frac{4}{5}$. Substitute to get $-\frac{4}{5} + \frac{3}{5} = \frac{-4+3}{5} = -\frac{1}{5}$.
• Evaluate $3a^2b$ when $a=\frac{1}{2}$ and $b=-\frac{1}{3}$. Substitute: $3(\frac{1}{2})^2(-\frac{1}{3}) = 3(\frac{1}{4})(-\frac{1}{3}) = -\frac{3}{12} = -\frac{1}{4}$.
• Evaluate $\frac{a+b}{c}$ when $a=-3$, $b=-5$, and $c=16$. Substitute to get $\frac{-3+(-5)}{16} = \frac{-8}{16} = -\frac{1}{2}$.

Explanation

This process combines algebra and fractions. Simply plug the given fraction values into the expression in place of the variables. After substituting, follow the order of operations to calculate the final answer, using your fraction arithmetic skills.