Fractions

Section 0.2 Fractions

Objective: Reduce, add, subtract, multiply, and divide with fractions.

Working with fractions is a very important foundation to algebra. Here we will briefly review reducing, multiplying, dividing, adding, and subtracting fractions. As this is a review, concepts will not be explained in detail as other lessons are.

World View Note: The earliest known use of fraction comes from the Middle Kingdom of Egypt around $2000$ BC!

We always like our final answers when working with fractions to be reduced. Reducing fractions is simply done by dividing both the numerator and denominator by the same number. This is shown in the following example

Example 0.2.1.

\begin{align*} \frac{36}{84} \amp \quad \text{Both numerator and denominator are divisible by } 4 \\ \frac{36\div 4}{84\div 4}=\frac{9}{21} \amp\quad \text{Both numerator and denominator are still divisible by } 3 \\ \frac{9\div 3}{21\div 3}=\frac{3}{7} \amp\quad \text{Our Solution } \end{align*}

The previous example could have been done in one step by dividing both numerator and denominator by $12\text{.}$ We also could have divided by $2$ twice and then divided by $3$ once (in any order). It is not important which method we use as long as we continue reducing our fraction until it cannot be reduced any further.

The easiest operation with fractions is multiplication. We can multiply fractions by multiplying straight across, multiplying numerators together and denominators together.

Example 0.2.2.

\begin{align*} \frac{6}{7}\cdot \frac{3}{5} \amp \quad \text{Multiply numerators across and denominators across } \\ \frac{18}{35} \amp\quad \text{Our Solution} \end{align*}

When multiplying we can reduce our fractions before we multiply. We can either reduce vertically with a single fraction, or diagonally with several fractions, as long as we use one number from the numerator and one number from the denominator.

Example 0.2.3.

\begin{align*} \frac{25}{24}\cdot\frac{32}{55} \amp \quad \text{Reduce $25$ and $55$ by dividing by $5$. Reduce $32$ and $24$ by dividing by } 8 \\ \frac{5}{3}\cdot\frac{4}{11} \amp\quad \text{Multiply numerators across and denominators across} \\ \frac{20}{33} \amp\quad \text{Our Solution} \end{align*}

Dividing fractions is very similar to multiplying with one extra step. Dividing fractions requires us to first take the reciprocal of the second fraction and multiply. Once we do this, the multiplication problem solves just as the previous problem

Example 0.2.4.

\begin{align*} \frac{21}{16}\div\frac{28}{6} \amp \quad \text{Multiply by the reciprocal } \\ \frac{21}{16}\cdot\frac{6}{28} \amp\quad \text{Reduce $21$ and $28$ by dividing by $7$. Reduce $6$ and $16$ by dividing by } 2 \\ \frac{3}{8}\cdot\frac{3}{4} \amp\quad \text{Multiply numerators across and denominators across} \\ \frac{9}{32} \amp\quad \text{Our Solution} \end{align*}

To add and subtract fractions we will first have to find the least common denominator (LCD). There are several ways to find an LCD. One way is to find the smallest multiple of the largest denominator that you can also divide the small denomiator by.

Example 0.2.5.

\begin{align*} 8 -3 \amp \quad \text{Add the opposite of } 3 \\ 8+(-3) \amp \quad \text{Different signs, subtract $8 - 3=5$, use sign from bigger number, positive} \\ 5 \amp\quad \text{Our Solution} \end{align*}

Example 0.2.6.

\begin{align*} \text{Find the LCD of and } 12 \amp \quad \text{Test multiples of } 12 \\ 12\text{? }\frac{12}{8} \amp \quad \text{Can't divide $12$ by $8$!} \\ 24\text{? }\frac{24}{8} \amp \quad \text{Yes! We can divide $24$ by $8$!} \\ 24 \amp\quad \text{Our Solution} \end{align*}

Adding and subtracting fractions is identical in process. If both fractions already have a common denominator we just add or subtract the numerators and keep the denominator.

Example 0.2.7.

\begin{align*} \frac{7}{8}+\frac{3}{8} \amp \quad \text{Same denominator, add numerators $7 +3$, keep common denominator } 8 \\ \frac{10}{8} \amp \quad \text{Reduce answer, dividing by } 2 \\ \frac{5}{4} \amp\quad \text{Our Solution} \end{align*}

While $\frac{5}{4}$ can be written as the mixed number $1\frac{1}{4}\text{,}$ in algebra we will almost never use mixed numbers. For this reason we will always use the improper fraction, not the mixed number.

Example 0.2.8.

\begin{align*} \frac{13}{6}-\frac{9}{6} \amp \quad \text{ Same denominator, subtract numerators $13 - 9$, keep common denominator} \\ \frac{4}{6} \amp\quad \text{Reduce answer, dividing by } 2 \\ \frac{2}{3} \amp\quad \text{Our Solution} \end{align*}

If the denominators do not match we will first have to identify the LCD and build up each fraction by multiplying the numerators and denominators by the same number so the denominator is built up to the LCD.

Example 0.2.9.

\begin{align*} \frac{5}{6}+\frac{4}{9} \amp \quad \text{LCD is } 18 \\ \frac{3\cdot 5}{3\cdot 6}+\frac{4\cdot 2}{9\cdot 2} \amp \quad \text{Multiply first fraction by $3$ and the second by } 2 \\ \frac{15}{18}+\frac{8}{18} \amp \quad \text{Same denominator, add numerators } 15+8 \\ \frac{23}{18} \amp\quad \text{Our Solution} \end{align*}

Example 0.2.10.

\begin{align*} \frac{2}{3}-\frac{1}{6} \amp \quad \text{LCD is } 6 \\ \frac{2\cdot 2}{2\cdot 3}-\frac{1}{6} \amp \quad \text{Multiply first fraction by $2$, the second already has a denominator of } 6 \\ \frac{4}{6}-\frac{1}{6} \amp \quad \text{Same denominator, subtract numerators, 4 } 4-1 \\ \frac{3}{6} \amp \quad \text{Reduce answer, dividing by } 3 \\ \frac{1}{2} \amp\quad \text{Our Solution} \end{align*}

Exercises Exercises - Fractions

Exercise Group.

Simplify each. Leave your answer as an improper fraction.

1.

$\frac{42}{12}$

2.

$\frac{25}{30}$

3.

$\frac{35}{25}$

4.

$\frac{24}{9}$

5.

$\frac{54}{36} $

6.

$\frac{30}{24} $

7.

$\frac{45}{36} $

8.

$\frac{36}{27} $

9.

$\frac{27}{18} $

10.

$\frac{48}{18} $

11.

$\frac{40}{16} $

12.

$\frac{48}{42} $

13.

$\frac{63}{18} $

14.

$\frac{16}{12} $

15.

$\frac{80}{60} $

16.

$\frac{72}{48} $

17.

$\frac{72}{60} $

18.

$\frac{126}{108} $

19.

$\frac{36}{24} $

20.

$\frac{160}{140} $

Exercise Group.

Find each product.

21.

$(9)\left(\frac{8}{9}\right) $

22.

$(-2)\left(-\frac{5}{6}\right) $

23.

$(2)\left(-\frac{2}{9}\right) $

24.

$(-2)\left(\frac{1}{3}\right) $

25.

$(-2)\left(\frac{13}{8}\right) $

26.

$\left(\frac{3}{2}\right)\left(\frac{1}{2}\right) $

27.

$\left(-\frac{6}{5}\right)\left(-\frac{11}{8}\right)$

28.

$\left(-\frac{3}{7}\right)\left(\frac{11}{8}\right)$

29.

$(8)\left(\frac{1}{2}\right)$

30.

$(-2)\left(-\frac{9}{7}\right)$

31.

$\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)$

32.

$\left(-\frac{17}{9}\right)\left(-\frac{3}{5}\right)$

33.

$(2)\left(\frac{3}{2}\right) $

34.

$\left(\frac{17}{9}\right)\left(-\frac{3}{5}\right)$

35.

$\left(\frac{1}{2}\right)\left(-\frac{7}{5}\right)$

36.

$\left(\frac{1}{2}\right)\left(\frac{5}{7}\right)$

Exercise Group.

Find each quotient.

37.

$-2\div \frac{7}{4}$

38.

$\frac{-12}{7}\div \frac{-9}{5}$

39.

$\frac{-1}{9}\div \frac{-1}{2}$

40.

$-2\div\frac{-3}{2}$

41.

$\frac{-3}{2}\div \frac{13}{7}$

42.

$\frac{5}{3}\div \frac{7}{5}$

43.

$-1\div\frac{2}{3}$

44.

$\frac{10}{9}\div(-6)$

45.

$\frac{8}{9}\div \frac{1}{5}$

46.

$\frac{1}{6}\div \frac{-5}{3}$

47.

$\frac{-9}{7}\div \frac{1}{5}$

48.

$\frac{-13}{8}\div \frac{-15}{8}$

49.

$\frac{-2}{9}\div \frac{-3}{2}$

50.

$\frac{-4}{5}\div \frac{-13}{8}$

51.

$\frac{1}{10}\div \frac{3}{2}$

52.

$\frac{5}{3}\div \frac{5}{3}$

Exercise Group.

Evaluate each expression.

53.

$\frac{1}{3}+\left(- \frac{4}{3}\right)$

54.

$\frac{1}{7}+\left(- \frac{11}{7}\right)$

55.

$\frac{3}{7}- \frac{1}{7}$

56.

$\frac{1}{3}+\frac{5}{3}$

57.

$\frac{11}{6}+\frac{7}{6}$

58.

$(-2) + \left(- \frac{15}{8}\right)$

59.

$\frac{3}{5}+ \frac{5}{4}$

60.

$(-1)- \frac{2}{3}$

61.

$\frac{2}{5}+\frac{5}{4}$

62.

$\frac{12}{7}- \frac{9}{7}$

63.

$\frac{9}{8}+\left(- \frac{2}{7}\right)$

64.

$(-2)+ \frac{5}{6} $

65.

$1+\left(- \frac{1}{3}\right)$

66.

$\frac{1}{2}-\left( \frac{11}{6}\right)$

67.

$\left(-\frac{1}{2}\right)+ \frac{3}{2}$

68.

$\frac{11}{8}- \frac{1}{2}$

69.

$\frac{1}{5}+ \frac{3}{4}$

70.

$\frac{6}{5}- \frac{8}{5}$

71.

$\left(-\frac{5}{7}\right)- \frac{15}{8}$

72.

$\left(-\frac{1}{3}\right)+\left(- \frac{8}{5}\right)$

73.

$6- \frac{8}{7}$

74.

$(-6)+\left(- \frac{5}{3}\right)$

75.

$\frac{3}{2}- \frac{15}{8}$

76.

$(-1)-\left(- \frac{1}{3}\right)$

77.

$\left(-\frac{15}{8}\right)+\frac{5}{3} $

78.

$\frac{3}{2}+ \frac{8}{7} $

79.

$(-1)-\left(- \frac{1}{6}\right)$

80.

$\left(-\frac{1}{2}\right)-\left(- \frac{3}{5}\right)$

81.

$\frac{5}{3}-\left(- \frac{1}{3}\right)$

82.

$\frac{9}{7}-\left(- \frac{5}{3}\right)$