3.6: Fractions and Decimals (2023)

Terminating Decimals

First reduce the fraction to lowest terms. If the denominator of the resulting fraction has a prime factorization consisting of strictly twos and/or fives, then the decimal representation will “terminate.”

Example 1

Change 15/48 to a decimal.

Solution

First, reduce the fraction to lowest terms.

\[ \begin{aligned} \frac{15}{48} = \frac{3 \cdot 5}{3 \cdot 16} \\ = \frac{5}{16} \end{aligned}\nonumber \]

Next, note that the denominator of 5/16 has prime factorization 16 = 2·2·2·2. It consists only of twos. Hence, the decimal representation of 5/16 should terminate.

The zero remainder terminates the process. Hence, 5/16 = 0.3125.

Exercise

Change 10/16 to a decimal.

Answer

0.625

Example 2

Change \(3 \frac{7}{20}\) to a decimal.

Solution

Note that 7/20 is reduced to lowest terms and its denominator has prime factorization 20 = 2 · 2 · 5. It consists only of twos and fives. Hence, the decimal representation of 7/20 should terminate.

The zero remainder terminates the process. Hence, 7/20 = 0.35. Therefore, \(3 \frac{7}{20}\) = 3.35.

Exercise

Change \(7 \frac{11}{20}\) to a decimal.

Answer

7.55

Repeating Decimals

First reduce the fraction to lowest terms. If the prime factorization of the resulting denominator does not consist strictly of twos and fives, then the division process will never have a remainder of zero. However, repeated patterns of digits must eventually reveal themselves.

Example 3

Change 1/12 to a decimal.

Solution

Note that 1/12 is reduced to lowest terms and the denominator has a prime factorization 12 = 2 · 2 · 3 that does not consist strictly of twos and fives. Hence, the decimal representation of 1/12 will not “terminate.” We need to carry out the division until a remainder reappears for a second time. This will indicate repetition is beginning.

Note the second appearance of 4 as a remainder in the division above. This is an indication that repetition is beginning. However, to be sure, let’s carry the division out for a couple more places.

Note how the remainder 4 repeats over and over. In the quotient, note how the digit 3 repeats over and over. It is pretty evident that if we were to carry out the division a few more places, we would get

\[\frac{1}{12} = 0.833333 \cdots\nonumber \]

The ellipsis is a symbolic way of saying that the threes will repeat forever. It is the mathematical equivalent of the word “etcetera.”

Exercise

Change 5/12 to a decimal.

Answer

0.41666...

Using the Repeating Bar

To use the repeating bar notation, take whatever block of digits are under the repeating bar and duplicate that block of digits infinitely to the right.

Important Observation

Although \(0.8 \overline{33}\) will also produce 0.8333333 ..., as a rule we should use as few digits as possible under the repeating bar. Thus, \(0.8 \overline{3}\) is preferred over \(0.8 \overline{33}\).

Example 4

Change 23/111 to a decimal.

Solution

The denominator of 23/111 has prime factorization 111 = 3 ·37 and does not consist strictly of twos and fives. Hence, the decimal representation will not “terminate.” We need to perform the division until we spot a repeated remainder.

Note the return of 23 as a remainder. Thus, the digit pattern in the quotient should start anew, but let’s add a few places more to our division to be sure.

Aha! Again a remainder of 23. Repetition! At this point, we are confident that

\[ \frac{23}{111} = 0.207207 ....\nonumber \]

Using a “repeating bar,” this result can be written

\[ \frac{23}{111} = 0. \overline{207}.\nonumber \]

Exercise

Change 5/33 to a decimal.

Answer

0.151515...

Example 5

Simplify: \(- \frac{3}{8} - 1.25\).

Solution

Let’s change 1.25 to an improper fraction.

\[ \begin{aligned} 1.25 = \frac{125}{100} ~ & \textcolor{red}{ \text{ Two decimal places } \Rightarrow \text{ two zeroes.} \\ = \frac{5}{4} ~ & \textcolor{red}{ \text{ Reduce to lowest terms.}} \end{aligned}\nonumber \]

In the original problem, replace 1.25 with 5/4, make equivalent fractions with a common denominator, then subtract.

\[ \begin{aligned} - \frac{3}{8} - 1.25 = - \frac{3}{8} - \frac{5}{4} ~ & \textcolor{red}{ \text{ Replace 1.25 with 5/4.}} \\ = - \frac{3}{8} - \frac{5 \cdot \textcolor{red}{2}}{4 \cdot \textcolor{red}{2}} ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 8.}} \\ = - \frac{3}{8} - \frac{10}{8} ~ & \textcolor{red}{ \text{ Simplify the numerator and denominator.}} \\ = - \frac{3}{8} + \left( - \frac{10}{8} \right) ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ = = \frac{13}{8} ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber \]

Thus, −3/8 − 1.25 = −13/8.

Alternate Solution

Because −3/8 is reduced to lowest terms and 8 = 2 ·2 ·2 consists only of twos, the decimal representation of −3/8 will terminate.

Hence, −3/8 = −0.375. Now, replace −3/8 in the original problem with −0.375, then simplify.

\[ \begin{aligned} - \frac{3}{8} - 1.25 = -0.375 - 1.25 ~ & \textcolor{red}{ \text{ Replace } -3/8 \text{ with } -0.375.} \\ =-0.375 + (-1.25) ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ = -1.625 ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber \]

Thus, −3/8 − 1.25 = −1.625.

Are They the Same?

The first method produced −13/8 as an answer; the second method produced −1.625. Are these the same results? One way to find out is to change −1.625 to an improper fraction.

\[ \begin{aligned} -1.625 = - \frac{1625}{1000} ~ & \textcolor{red}{ \text{ Three places } \Rightarrow \text{ three zeroes.}} \\ = - \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot 13}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} ~ & \textcolor{red}{ \text{ Prime factor.}} \\ = - \frac{13}{2 \cdot 2 \cdot 2} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ = - \frac{13}{8} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]

Thus, the two answers are the same.

Exercise

Simplify: \(− \frac{7}{8} − 6.5\)

Answer

\(−7 \frac{3}{8}\) or −7.375

Example 6

Simplify: \(-\frac{2}{3} + 0.35\).

Solution

Let’s attack this expression by first changing 0.35 to a fraction.

\[ \begin{aligned} - \frac{2}{3} + 0.35 = - \frac{2}{3} + \frac{35}{100} ~ & \textcolor{red}{ \text{ Change 0.35 to a fraction.}} \\ = - \frac{2}{3} + \frac{7}{20} ~ & \textcolor{red}{ \text{ Reduce 35/100 to lowest terms.}} \end{aligned}\nonumber \]

Find an LCD, make equivalent fractions, then add.

\[ \begin{aligned} = - \frac{2 \cdot 20}{3 \cdot 20} + \frac{7 \cdot 3}{20 \cdot 3} ~ & \textcolor{red}{ \text{ Equivalent fractions with LCD = 60.}} \\ = - \frac{40}{60} + \frac{21}{60} ~ & \textcolor{red}{ \text{ Simplify numerators and denominators.}} \\ = - \frac{19}{60} ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber \]

Then, \(-\frac{2}{3} +0.35 = - \frac{19}{60}\).

Exercise

Simplify: \(-\frac{4}{9} + 0.25\)

Answer

−7/36

Important Observation

When presented with a problem containing both decimals and fractions, if the decimal representation of any fraction repeats, its best to first change all numbers to fractions, then simplify.