6.8: Converting a Fraction to a Decimal

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Learning Objectives

be able to convert a fraction to a decimal

Now that we have studied and practiced dividing with decimals, we are also able to convert a fraction to a decimal. To do so we need only recall that a fraction bar can also be a division symbol. Thus, $\dfrac{3}{4}$ not only means "3 objects out of 4," but can also mean "3 divided by 4."

Sample Set A

Convert the following fractions to decimals. If the division is nonterminating, round to two decimal places.

$\dfrac{3}{4}$. Divide 3 by 4.

Solution

$\begin{array} {.75} \\ {4\overline{)3.00}} \\ {\underline{2.8\ \ }} \\ {20} \\ {\underline{20}} \\ {0} \end{array}$

Thus, $\dfrac{3}{4} = 0.75$.

Sample Set A

$\dfrac{1}{5}$. Divide 1 by 5.

Solution

$\begin{array} {.2} \\ {5\overline{)1.0}} \\ {\underline{1.0}} \\ {0} \end{array}$

Thus, $\dfrac{1}{5} = 0.2$.

Sample Set A

$\dfrac{5}{6}$. Divide 5 by 6.

Solution

$Long division. 5 divided by 6 ends in a recurring remainder. The quotient is .833. The recurring remainder indicates that the division is nonterminating.$

$\dfrac{5}{6} = 0.833...$ We are to round to two decimal places.

Thus, $\dfrac{5}{6} = 0.83$ to two decimal places.

Sample Set A

$5 \dfrac{1}{8}$. Note that $5 \dfrac{1}{8} = 5 + \dfrac{1}{8}$

Solution

Convert $\dfrac{1}{8}$ to a decimal.

$\begin{array} {r} {.125} \\ {8\overline{)1.000}} \\ {\underline{\ \ \ 8\ \ \ \ }} \\ {20\ \ } \\ {\underline{16\ \ }} \\ {40} \\ {\underline{40}} \\ {0} \end{array}$

$\dfrac{1}{8} = .125$

Thus, $5 \dfrac{1}{8} = 5 + \dfrac{1}{8} = 5 + .125 = 5.125$.

Sample Set A

$0.16 \dfrac{1}[4}$. This is a complex decimal.

Solution

Note that the 6 is in the hundredths position.

The number $0.16\dfrac{1}{4}$ is read as "sixteen and one-fourth hundredths."

$0.16 \dfrac{1}{4} = \dfrac{16\dfrac{1}{4}}{100} = \dfrac{\dfrac{16 \cdot 4 + 1}{4}}{100} = \dfrac{\dfrac{65}{4}}{\dfrac{100}{1}} = \dfrac{\begin{array} {c} {^{13}} \\ {\cancel{65}} \end{array}}{4} \cdot \dfrac{1}{\begin{array} {c} {\cancel{100}} \\ {^{20}} \end{array}} = \dfrac{13 \cdot 1}{4 \cdot 20} = \dfrac{13}{80}.$

Now, convert $\dfrac{13}{80}$ to a decimal.

$\begin{array} {r} {.1625} \\ {80\overline{)13.0000}} \\ {\underline{8\ 0\ \ \ \ \ \ }} \\ {5\ 00\ \ \ \ } \\ {\underline{4\ 80\ \ \ \ }} \\ {200\ \ } \\ {\underline{160\ \ }} \\ {400} \\ {\underline{400}} \\ {0} \end{array}$

Thus, $0.16 \dfrac{1}{4} = 0.1625$.

Practice Set A

Convert the following fractions and complex decimals to decimals (in which no proper fractions appear). If the divison is nonterminating, round to two decimal places.

$\dfrac{1}{4}$

Answer: 0.25

Practice Set A

$\dfrac{1}{25}$

Answer: 0.04

Practice Set A

$\dfrac{1}{6}$

Answer: 0.17

Practice Set A

$\dfrac{15}{16}$

Answer: 0.9375

Practice Set A

$0.9 \dfrac{1}{2}$

Answer: 0.95

Practice Set A

$8.0126 \dfrac{3}{8}$

Answer: 8.0126375

Exercises

For the following 30 problems, convert each fraction or complex decimal number to a decimal (in which no proper fractions appear).

Exercise $\PageIndex{1}$

$\dfrac{1}{2}$

Answer: 0.5

Exercise $\PageIndex{2}$

$\dfrac{4}{5}$

Exercise $\PageIndex{3}$

$\dfrac{7}{8}$

Answer: 0.875

Exercise $\PageIndex{4}$

$\dfrac{5}{8}$

Exercise $\PageIndex{5}$

$\dfrac{3}{5}$

Answer: 0.6

Exercise $\PageIndex{6}$

$\dfrac{2}{5}$

Exercise $\PageIndex{7}$

$\dfrac{1}{25}$

Answer: 0.04

Exercise $\PageIndex{8}$

$\dfrac{3}{25}$

Exercise $\PageIndex{9}$

$\dfrac{1}{20}$

Answer: 0.05

Exercise $\PageIndex{10}$

$\dfrac{1}{15}$

Exercise $\PageIndex{11}$

$\dfrac{1}{50}$

Answer: 0.02

Exercise $\PageIndex{12}$

$\dfrac{1}{75}$

Exercise $\PageIndex{13}$

$\dfrac{1}{3}$

Answer: $0.\overline{3}$

Exercise $\PageIndex{14}$

$\dfrac{5}{6}$

Exercise $\PageIndex{15}$

$\dfrac{3}{16}$

Answer: 0.1875

Exercise $\PageIndex{16}$

$\dfrac{9}{16}$

Exercise $\PageIndex{17}$

$\dfrac{1}{27}$

Answer: $0.0\overline{37}$

Exercise $\PageIndex{18}$

$\dfrac{5}{27}$

Exercise $\PageIndex{19}$

$\dfrac{7}{13}$

Answer: $0.\overline{538461}$

Exercise $\PageIndex{20}$

$\dfrac{9}{14}$

Exercise $\PageIndex{21}$

$7 \dfrac{2}{3}$

Answer: $7.\overline{6}$

Exercise $\PageIndex{22}$

$8\dfrac{5}{16}$

Exercise $\PageIndex{23}$

$1 \dfrac{2}{15}$

Answer: $1.1\overline{3}$

Exercise $\PageIndex{24}$

$65\dfrac{5}{22}$

Exercise $\PageIndex{25}$

$101 \dfrac{6}{25}$

Answer: 101.24

Exercise $\PageIndex{26}$

$0.1 \dfrac{1}{2}$

Exercise $\PageIndex{27}$

$0.24\dfrac{1}{8}$

Answer: 0.24125

Exercise $\PageIndex{28}$

$5.66 \dfrac{2}{3}$

Exercise $\PageIndex{29}$

$810.3106 \dfrac{5}{16}$

Answer: 810.31063125

Exercise $\PageIndex{30}$

$4.1 \dfrac{1}{9}$

For the following 18 problems, convert each fraction to a decimal. Round to five decimal places.

Exercise $\PageIndex{31}$

$\dfrac{1}{9}$

Answer: 0.11111

Exercise $\PageIndex{32}$

$\dfrac{2}{9}$

Exercise $\PageIndex{33}$

$\dfrac{3}{9}$

Answer: 0.33333

Exercise $\PageIndex{34}$

$\dfrac{4}{9}$

Exercise $\PageIndex{35}$

$\dfrac{5}{9}$

Answer: 0.55556

Exercise $\PageIndex{36}$

$\dfrac{6}{9}$

Exercise $\PageIndex{37}$

$\dfrac{7}{9}$

Answer: 0.77778

Exercise $\PageIndex{38}$

$\dfrac{8}{9}$

Exercise $\PageIndex{39}$

$\dfrac{1}{11}$

Answer: 0.09091

Exercise $\PageIndex{40}$

$\dfrac{2}{11}$

Exercise $\PageIndex{41}$

$\dfrac{3}{11}$

Answer: 0.27273

Exercise $\PageIndex{42}$

$\dfrac{4}{11}$

Exercise $\PageIndex{43}$

$\dfrac{5}{11}$

Answer: 0.45455

Exercise $\PageIndex{44}$

$\dfrac{6}{11}$

Exercise $\PageIndex{45}$

$\dfrac{7}{11}$

Answer: 0.63636

Exercise $\PageIndex{46}$

$\dfrac{8}{11}$

Exercise $\PageIndex{47}$

$\dfrac{9}{11}$

Answer: 0.81818

Exercise $\PageIndex{48}$

$\dfrac{10}{11}$

Calculator Problems
For the following problems, use a calculator to convert each fraction to a decimal. If no repeating pattern seems to exist, round to four decimal places.

Exercise $\PageIndex{49}$

$\dfrac{16}{125}$

Answer: 0.128

Exercise $\PageIndex{50}$

$\dfrac{85}{311}$

Exercise $\PageIndex{51}$

$\dfrac{192}{197}$

Answer: 0.9746

Exercise $\PageIndex{52}$

$\dfrac{1}{1469}$

Exercise $\PageIndex{53}$

$\dfrac{4}{21,015}$

Answer: 0.0002

Exercise $\PageIndex{54}$

$\dfrac{81,426}{106,001}$

Exercise $\PageIndex{55}$

$\dfrac{16,501}{426}$

Answer: 38.7347

Exercises for Review

Exercise $\PageIndex{56}$

Round 2,105,106 to the nearest hundred thousand.

Exercise $\PageIndex{57}$

$\dfrac{8}{5}$ of what number is $\dfrac{3}{2}$?

Answer: $\dfrac{15}{16}$

Exercise $\PageIndex{58}$

Arrange $1 \dfrac{9}{16}$, $1 \dfrac{5}{8}$, and $1 \dfrac{7}{12}$ in increasing order.

Exercise $\PageIndex{59}$

Convert the complex decimal $3.6 \dfrac{5}{4}$ to a fraction.

Answer: $3 \dfrac{29}{40}$ or 3.725

Exercise $\PageIndex{60}$

Find the quotient. $30 \div 1.1$.