0.2: Fractions

Example \(\PageIndex{1}\)

Simplify: \(\dfrac{36}{84}\)

Solution

\[\begin{array}{rl} \dfrac{36}{84} & \text{Divide by a common factor of 4} \\ & & \\ \dfrac{36 \div 4}{84 \div 4} = \dfrac{9}{21} & \text{Divide by a common factor of 3}\\ && \\ \dfrac{9 \div 3}{21 \div 3} = \dfrac{3}{7} & \text{No more common factors} \\&& \\ \dfrac{3}{7} & \text{Simplified fraction} \end{array}\nonumber\]

Example \(\PageIndex{2}\)

Multiply: \(\dfrac{6}{7} \cdot \dfrac{3}{5}\)

Solution

\[\begin{array}{rl} \dfrac{6}{7} \cdot \dfrac{3}{5} & \text{Multiply across numerators and denominators}\\ & & \\ \dfrac{6 \cdot 3}{7 \cdot 5} & \text{Simplify} \\ && \\ \dfrac{18}{35} & \text{No common factors} \\ &&\\ \dfrac{18}{35} & \text{Product} \end{array}\nonumber\]

Example \(\PageIndex{3}\)

Multiply: \(\dfrac{25}{24} \cdot \dfrac{32}{55}\)

Solution

Let’s reduce each fraction first, then multiply. \[\begin{array}{rl} \dfrac{25}{24} \cdot \dfrac{32}{55} & \text{Reduce 25 & 55 by a common factor of 5} \\ & & \\ \dfrac{5}{24} \cdot \dfrac{32}{11} & \text{Reduce 24 & 32 by a common factor of 8} \\ & & \\ \dfrac{5}{3} \cdot \dfrac{4}{11} & \text{Multiply fractions} \\ & & \\ \dfrac{20}{33} & \text{No common factors} \\ && \\ \dfrac{20}{33} & \text{ Product} \end{array}\nonumber\]

Example \(\PageIndex{4}\)

Multiply: \(\dfrac{25}{24} \cdot \dfrac{32}{55}\)

Solution

Let’s multiply first, then reduce the fraction. \[\begin{array}{rl} \dfrac{5}{6} \cdot \dfrac{3}{10} & \text{Multiply fractions} \\ & & \\ \dfrac{15}{60} & \text{Reduce by a factor of 15} \\ & & \\ \dfrac{15 \div 15}{60 \div 15} & \text{Simplify} \\ & & \\ \dfrac{1}{4} & \text{No common factors} \\ && \\ \dfrac{1}{4} & \text{ Product} \end{array}\nonumber\]

Example \(\PageIndex{5}\)

Divide: \(\dfrac{21}{16} \div \dfrac{28}{6}\)

Solution

\[\begin{array}{rl} \dfrac{21}{16} \div \dfrac{28}{6} & \text{Rewrite the expression as a product} \\ & & \\ \dfrac{21}{16} \textcolor{blue}{\cdot \dfrac{6}{28}} & \text{Reduce the fractions} \\ & & \\ \dfrac{3}{8} \cdot \dfrac{3}{4} & \text{Multiply fractions}\\ & & \\ \dfrac{9}{32} & \text{Quotient} \end{array}\nonumber\]

Example \(\PageIndex{6}\)

Divide: \(\dfrac{\dfrac{14}{15}}{\dfrac{7}{60}}\)

Solution

\[\begin{array}{rl} \dfrac{\dfrac{14}{15}}{\dfrac{7}{60}} & \text{Rewrite the complex fraction with the division sign} \\ && \\ \dfrac{14}{15} \div \dfrac{7}{60} & \text{Rewrite the expression as a product} \\ & & \\ \dfrac{14}{15} \textcolor{blue}{\cdot \dfrac{60}{7}} & \text{Reduce the fractions} \\ & & \\ \dfrac{2}{1} \cdot \dfrac{4}{1} & \text{Multiply fractions}\\ & & \\ \dfrac{8}{1} & \text{Simplify} \\ && \\ 8 & \text{Quotient} \end{array}\nonumber\]

Example \(\PageIndex{7}\)

Find LCM\((2,3,5)\).

Solution

We need to think of a multiple of 2, 3, and 5 that is divisible by these numbers. If we multiply 2, 3, and 5, we get \[2 \cdot 3 \cdot 5 = 30\nonumber \] And so, the LCM\((2,3,5) = 30\) because 30 is divisible by 2, 3, and 5.

Example \(\PageIndex{8}\)

Find LCM\((6,35,54)\).

Solution

When the numbers aren’t as obvious, then we can use the strategy below to find the LCM:

Step 1. Find the prime factorization of each number in your set. \[\begin{aligned} 6 &= 2 \cdot 3 \\ 35 &= 5 \cdot 7 \\ 54 &= 2 \cdot 3^3 \\ \end{aligned}\]

Step 2. Look at all the factors and take one of each factor. For the factors with exponents, take the factors with the highest exponent. \[\begin{array}{rl} 2 & \text{take 2} \\ 3^3 & \text{take 3 with the highest exponent} \\ 5 & \text{take 5} \\ 7 & \text{take 7} \\ \end{array}\nonumber\]

Step 3. Multiply the numbers found in the previous step. This product is the LCM. \[\text{LCM}(6,35,54) = 2 \cdot 3^3 \cdot 5 \cdot 7 = 1890\nonumber \]

Example \(\PageIndex{9}\)

Find the LCD between \(\dfrac{5}{6}\) and \(\dfrac{4}{9}\). Rewrite each fraction with the LCD.

Solution

If we need to obtain the LCD, then we can follow a series of steps.

Step 1. Find the LCD, i.e., the LCM between denominators. In this case, we need to find the LCM\((6,9)\). \[\begin{aligned} 6 &= 2 \cdot 3 \\ 9 &= 3^2 \\ \end{aligned}\] We can see that the LCM\((6,9) = 2 \cdot 3^2 = 18\). This is the LCD.

Step 2. Next, we rewrite each fraction with the LCD. \[\begin{array}{rl} \dfrac{5}{6} & \text{Multiply the numerator and denominator by 3} \\ \dfrac{5}{6} \cdot \dfrac{3}{3} & \text{Notice we get 18 in the denominator} \\ \dfrac{15}{18} & \text{The denominator is the LCD} \checkmark \\\end{array}\nonumber\] \[\begin{array}{rl} \dfrac{4}{9} & \text{Multiply the numerator and denominator by 2} \\ \dfrac{4}{9} \cdot \dfrac{2}{2} & \text{Notice we get 18 in the denominator} \\ \dfrac{8}{18} & \text{The denominator is the LCD} \checkmark \\\end{array}\nonumber\]

Example \(\PageIndex{10}\)

Add: \(\dfrac{7}{8} + \dfrac{3}{8}\)

Solution

\[\begin{array}{rl} \dfrac{7}{8} + \dfrac{3}{8} & \text{Same denomintaor, add across numerators} \\ & & \\ \dfrac{10}{8} & \text{Reduce by a common factor of 2} \\ & & \\ \dfrac{5}{4} & \text{Sum} \end{array}\nonumber\]

Example \(\PageIndex{11}\)

Subtract: \(\dfrac{13}{6} - \dfrac{9}{6}\)

Solution

\[\begin{array}{rl} \dfrac{13}{6} - \dfrac{9}{6} & \text{Same denomintaor, subtract across numerators} \\ & & \\ \dfrac{4}{6} & \text{Reduce by a common factor of 2} \\ & & \\ \dfrac{2}{3} & \text{Difference} \end{array}\nonumber\]

Example \(\PageIndex{12}\)

Add: \(\dfrac{5}{6} + \dfrac{4}{9}\)

Solution

\[\begin{array}{rl} \dfrac{5}{6} + \dfrac{4}{9} & \text{Unlike denominators; LCD}(6,9) = 18 \\ & & \\ \dfrac{5}{6} \cdot \dfrac{3}{3}+ \dfrac{4}{9} \cdot \dfrac{2}{2} & \text{Rewrite each fraction with the LCD} \\ & & \\ \dfrac{15}{18} + \dfrac{8}{18} & \text{Same denominator, add across numerators} \\ & & \\ \dfrac{23}{18} & \text{No common factors} \\ & & \\ \dfrac{23}{18} & \text{Sum} \end{array}\nonumber\]

Example \(\PageIndex{13}\)

Subtract: \(\dfrac{2}{3} - \dfrac{1}{6}\)

Solution

\[\begin{array}{rl} \dfrac{2}{3} - \dfrac{1}{6} & \text{Unlike denominators; LCD}(3,6) = 6 \\ & & \\ \dfrac{2}{3} \cdot \dfrac{2}{2}+ \dfrac{1}{6} & \text{Rewrite each fraction with the LCD} \\ & & \\ \dfrac{4}{6} - \dfrac{1}{6} & \text{Same denominator, subtract across numerators} \\ & & \\ \dfrac{3}{6} & \text{Reduce by a common factor of 3} \\ & & \\ \dfrac{1}{2} & \text{Difference} \end{array}\nonumber\]