Add and Subtract Fractions with Different Denominators

Examples

Answers

EXAMPLE 4.67

	\(\dfrac{1}{2}+\dfrac{1}{3}\)
Find the LCD of 2, 3. \[ \begin{aligned} 2 &=2 \\ 3 &=3 \\ \hline \mathrm{LCD} &=2 \cdot 3 \\ \mathrm{LCD} &=6 \end{aligned} \]	\(\dfrac{1 \cdot 3}{2 \cdot 3}+\dfrac{1 \cdot 2}{3 \cdot 2}\)
Change into equivalent fractions with the LCD 6.	\(\dfrac{1 \cdot 3}{2 \cdot 3}+\dfrac{1 \cdot 2}{3 \cdot 2}\)
Simplify the numerators and denominators.	\(\dfrac{3}{6}+\dfrac{2}{6}\)
Add.	\(\dfrac{5}{6}\)

Remember, always check to see if the answer can be simplified. Since \(5\) and \(6\) have no common factors, the fraction \(\frac{5}{6}\) cannot be reduced.

EXAMPLE 4.68

	\(\dfrac{1}{2}-\left(-\dfrac{1}{4}\right)\)
Find the LCD of 2 and 4. \[ \begin{aligned} 2 &=2 \\ 4 &=2 \cdot 2 \\ \hline \mathrm{LCD} &=2 \cdot 2 \\ \mathrm{LCD} &=4 \end{aligned} \]
Rewrite as equivalent fractions using the LCD 4.	\(\dfrac{1 \cdot 2}{2 \cdot 2}-\left(-\dfrac{1}{4}\right)\)
Simplify the first fraction.	\(\dfrac{2}{4}-\left(-\dfrac{1}{4}\right)\)
Subtract.	\(\dfrac{2-(-1)}{4}\)
Simplify.	\(\dfrac{3}{4}\)

One of the fractions already had the least common denominator, so we only had to convert the other fraction.

EXAMPLE 4.69

	\(\dfrac{7}{12}+\dfrac{5}{18}\)
Find the LCD of 12 and 18. \[ \begin{aligned} 12 &=2 \cdot 2 \cdot 3 \\ 18 &=2 \cdot 3 \cdot 3 \\ \hline \mathrm{LCD} &=2 \cdot 2 \cdot 3 \cdot 3 \\ \mathrm{LCD} &=36 \end{aligned} \]
Rewrite as equivalent fractions with the LCD.	\(\dfrac{7 \cdot 3}{12 \cdot 3}+\dfrac{5 \cdot 2}{18 \cdot 2}\)
Simplify the numerators and denominators.	\(\dfrac{21}{36}+\dfrac{10}{36}\)
Add.	\(\dfrac{31}{36}\)

Because \(31\) is a prime number, it has no factors in common with \(36\). The answer is simplified.

EXAMPLE 4.70

	\(\dfrac{7}{15}-\dfrac{19}{24}\)
Find the LCD. \[ \begin{aligned} &15=\\ &24=2 \cdot 2 \cdot 2 \cdot 3\\ \hline &\begin{aligned} &\mathrm{LCD}=2 \cdot 2 \\ &\mathrm{LCD}=120 \end{aligned} \end{aligned} \] \(15\) is 'missing' three factors of \(2\) \(24\) is 'missing' a factor of \(5\)
Rewrite as equivalent fractions with the LCD.	\(\dfrac{7 \cdot 8}{15 \cdot 8}-\dfrac{19 \cdot 5}{24 \cdot 5}\)
Simplify each numerator and denominator.	\(\dfrac{56}{120}-\dfrac{95}{120}\)
Subtract.	\(-\dfrac{39}{120}\)
Rewrite showing the common factor of 3.	\(-\dfrac{13 \cdot 3}{40 \cdot 3}\)
Remove the common factor to simplify.	\(-\dfrac{13}{40}\)

EXAMPLE 4.71

	\(-\dfrac{11}{30}+\dfrac{23}{42}\)
Find the LCD. \(\begin{aligned} 30 &=2 \cdot 3 \cdot 5 \\ 42 &=2 \cdot 3 \cdot 7 \\ \hline \mathrm{LCD} &=2 \cdot 3 \cdot 5 \cdot 7 \\ \mathrm{LCD} &=210 \end{aligned}\)
Rewrite as equivalent fractions with the LCD.	\(-\dfrac{11 \cdot 7}{30 \cdot 7}+\dfrac{23 \cdot 5}{42 \cdot 5}\)
Simplify each numerator and denominator.	\(-\dfrac{77}{210}+\dfrac{115}{210}\)
Add.	\(\dfrac{38}{210}\)
Rewrite showing the common factor of 2.	\(\dfrac{19 \cdot 2}{105 \cdot 2}\)
Remove the common factor to simplify.	\(\dfrac{19}{105}\)

EXAMPLE 4.72

The fractions have different denominators.

	\(\dfrac{3}{5}+\dfrac{x}{8}\)
Find the LCD. \[ \begin{aligned} 5 &=\\ 8 &=2 \cdot 2 \cdot 2 \\ \hline \mathrm{LCD} &=2 \cdot 2 \cdot 2 \cdot 5 \\ \mathrm{LCD} &=40 \end{aligned} \]
Rewrite as equivalent fractions with the LCD.	\(\dfrac{3 \cdot 8}{5 \cdot 8}+\dfrac{x \cdot 5}{8 \cdot 5}\)
Simplify the numerators and denominators.	\(\dfrac{24}{40}+\dfrac{5 x}{40}\)
Add.	\(\dfrac{24+5 x}{40}\)

We cannot add \(24\) and \(5x\) since they are not like terms, so we cannot simplify the expression any further.