Unit 5: Common Fractions & Decimals
Topic A: Common Fractions & Decimals
The amount represented by a fraction may be expressed as a common fraction, a decimal, or as a percent.
We choose common fractions, decimals, or percents for convenience and to fit the standard way of doing things.
Common fractions are used:
- For everyday conversation about parts of the whole thing: [latex]\tfrac{1}{2}[/latex] cup of coffee, [latex]\tfrac{1}{4}[/latex] of an hour, [latex]\tfrac{3}{4}[/latex] tank of gas
- With amounts in the Imperial System of measurement, which is standard in the United States and still used by some people in Canada: [latex]3\tfrac{1}{4}[/latex] feet, [latex]\tfrac{5}{8}[/latex] inches, [latex]12\tfrac{3}{4}[/latex] miles, [latex]6\tfrac{1}{4}[/latex] pounds, [latex]1\tfrac{1}{2}[/latex] teaspoons
- For stock market reports and stock values
- For the score on the top of a test (which is usually changed to a percent)
Decimals are used
- With money ($12.23)
- With the metric system of measurement (1.5 metres, 7.25 litres, 29.75 kilometres, 0.5
centimetres, 9.2 grams, 75.5 kilograms, etc.) - Whenever there is a lot of arithmetic calculation to be done
- For calculators and computers
Percents are used
- For reporting statistics
- For bank rates and interest charges such as mortgage rates for reading a grade on a test
Writing Decimals as Common Fractions
Remember this skill?
- [latex]0.48 =\tfrac{48}{100}[/latex]
- [latex]3.542=3\tfrac{542}{1000}[/latex]
Common fractions should always be in lowest terms.
- [latex]0.48 =\tfrac{48}{100}\left(\tfrac{ ÷ 4}{ ÷ 4}\right)=\tfrac{12}{25}[/latex]
- [latex]3.542=3\tfrac{542}{1000}\left(\tfrac{ ÷ 2}{ ÷ 2}\right)=3\tfrac{271}{500}[/latex]
This list of factors may help you to simplify the fractions.
The factors of 10 are 1, 2, 5, 10
The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100
The factors of 1000 are 1, 2, 5, 8, 10, 20, 25, 50, 100, 125, 200, 250, 500, 1000
Remember: the whole number in a mixed decimal stays a whole number in a mixed fraction.
Exercise 1
Write these decimals as common fractions expressed in
lowest terms.
- [latex]16.04 =16\tfrac{4}{100}\left(\tfrac{÷4}{÷4}\right)=16\tfrac{1}{3}[/latex]
- [latex]0.085 =\tfrac{85}{1000}\left(\tfrac{÷5}{÷5}\right)=\tfrac{17}{200}[/latex]
- 3.48 =
- 12.075 =
- 6.25 =
- 0.14 =
- 12.125 =
- 1.75 =
- [latex]3\dfrac{12}{25}[/latex]
- [latex]12\dfrac{3}{40}[/latex]
- [latex]6\dfrac{1}{4}[/latex]
- [latex]\dfrac{7}{50}[/latex]
- [latex]12\dfrac{1}{8}[/latex]
- [latex]1\dfrac{3}{4}[/latex]
Some Tricky Conversions
Do you remember that there are some fractions that do not convert into decimals perfectly? The reason they do not is because they have a repeating decimal.
Some are:
- [latex]\dfrac{1}{6} = 0.1\overline{6}[/latex]
- [latex]\dfrac{1}{3} = 0.\overline{3}[/latex]
- [latex]\dfrac{2}{3} = 0.\overline{6}[/latex]
- [latex]\dfrac{5}{6}= 0.8\overline{3}[/latex]
- [latex]\dfrac{1}{9} = 0.\overline{1}[/latex]
It is not possible to convert [latex]\tfrac{1}{6}[/latex] (which is really [latex]0.166666666666\overline{6}[/latex]) into a fraction.
One way to deal with this problem is to memorize common ones like the above examples.
Writing Common Fractions as Decimals
As you know, common fractions with denominators of 10, 100, 1 000, or 10 000 are easily written as decimals.
- [latex]\dfrac{3}{10} = 0.3[/latex]
- [latex]\dfrac{21}{100} = 0.21[/latex]
- [latex]\dfrac{69}{1000} = 0.069[/latex]
But if the denominator is not a 10, 100, etc., you may be able to change a common fraction to an equivalent fraction with a denominator of 10, 100, 1000, or 10000 which can then be written easily as a decimal. For example,
- [latex]\dfrac{3}{5}\left(\dfrac{\times2}{\times2}\right)=\dfrac{6}{10} = 0.6[/latex]
- [latex]\dfrac{1}{2}\left(\dfrac{\times2}{\times2}\right)= \dfrac{5}{10} = 0.5[/latex]
- [latex]\dfrac{4}{25}\left(\dfrac{\times2}{\times2}\right)= \dfrac{16}{100} = 0.16[/latex]
Exercise 2
Write as decimals.
- [latex]\dfrac{1}{2}=\dfrac{5}{10}=0.5[/latex]
- [latex]\dfrac{2}{5}[/latex]
- [latex]\dfrac{7}{10}[/latex]
- [latex]\dfrac{4}{5}[/latex]
- [latex]\dfrac{21}{1000}[/latex]
- [latex]\dfrac{8}{25}[/latex]
Answers to Exercise 2
- 0.4
- 0.7
- 0.8
- 0.021
- 0.32
The following is a review of how to change a fraction to a decimal when it is not easy to make the denominator:
The line in a common fraction can be thought of as a divided by sign ÷
To change a common fraction to a decimal, do this:
numerator ÷ denominator = the decimal equivalent
Example A
[latex]\dfrac{3}{4}[/latex] Think [latex]3 ÷ 4[/latex]
- [latex]\begin{array} {r}0.75\\ \ 4\enclose{longdiv}{3.00}\\28\downarrow\\\hline 20 \hspace{0.1em} \\ 20\hspace{0.1em} \\ \hline\end{array}[/latex]
- [latex]\dfrac{3}{4} = 0.75[/latex]
Example B
[latex]\dfrac{3}{8}[/latex] Think [latex]3 ÷ 8[/latex]
- [latex]\begin{array}{r}0.375\\ 8\enclose{longdiv}{3.000}\\24\downarrow\hspace{0.7em}\\\hline60\hspace{0.8em}\\56\downarrow\\\hline40\hspace{0.1em}\\40\hspace{0.1em}\\\hline\end{array}[/latex]
- [latex]\dfrac{3}{8} = 0.375[/latex]
Example C
[latex]\dfrac{1}{3}[/latex] Think [latex]1 ÷ 3[/latex]
- [latex]\begin{array}{r}0.333\\ 3\enclose{longdiv}{1.000}\\{9}\downarrow\hspace{0.5em}\\\hline10\hspace{0.5em}\\0\downarrow\hspace{0.1em}\\\hline10\end{array}[/latex]
- [latex]\dfrac{1}{3}= 0.333[/latex]
Exercise 3
Use the division method to write these common fractions as decimals.
- [latex]\dfrac{1}{2}[/latex] = [latex]\begin{array}{r}0.5\\ 2\enclose{longdiv}{1.0}\\-1.0\\ \hline 0\end{array}[/latex]
[latex]\dfrac{1}{2} = 0.5[/latex] - [latex]\dfrac{1}{4}[/latex]
- [latex]\dfrac{2}{5}[/latex]
- [latex]\dfrac{6}{12}[/latex]
- [latex]\dfrac{1}{8}[/latex]
- [latex]\dfrac{3}{8}[/latex]
- [latex]\dfrac{2}{3}[/latex]
- [latex]\dfrac{19}{20}[/latex]
Answers to Exercise 3
- 0.25
- 0.4
- 0.5
- 0.125
- 0.375
- 0.6
- 0.95
The whole number in a mixed fraction stays a whole number in a mixed decimal. Rewrite the whole number to the left of the decimal. Then change the common fraction to a decimal.
- [latex]4 \dfrac{3}{4} = 4.75[/latex]
- Think [latex]\tfrac{3}{4} = 3 \div 4 = 0.75[/latex]
- [latex]16 \dfrac{1}{2} = 16.5[/latex]
- Think [latex]\tfrac{1}{2} = 1 \div 2 = 0.5[/latex]
Exercise 4
Complete the chart of equivalent common fractions and decimals. Use this chart as a reference for yourself in later work. Look for patterns that develop and note them in the margin.
| Common Fraction | Decimal |
|---|---|
| [latex]\dfrac{1}{8}[/latex] | |
| [latex]\dfrac{2}{8}=\dfrac{1}{4}[/latex] | |
| [latex]\dfrac{3}{8}[/latex] | |
| [latex]\dfrac{4}{8}=\dfrac{2}{4}=\dfrac{1}{2}[/latex] | |
| [latex]\dfrac{5}{8}[/latex] | |
| [latex]\dfrac{6}{8}=\dfrac{3}{4}[/latex] | |
| [latex]\dfrac{7}{8}[/latex] | |
| [latex]\dfrac{8}{8}=\dfrac{4}{4}=\dfrac{2}{2}=1[/latex] |
|
| [latex]\dfrac{1}{12}[/latex] | |
| [latex]\dfrac{2}{12}=\dfrac{1}{6}[/latex] | |
| [latex]\dfrac{4}{12}=\dfrac{2}{6}=\dfrac{1}{3}[/latex] | |
| [latex]\dfrac{6}{12}=\dfrac{3}{6}=\dfrac{1}{2}[/latex] | |
| [latex]\dfrac{8}{12}=\dfrac{4}{6}=\dfrac{2}{3}[/latex] | |
| [latex]\dfrac{10}{12}=\dfrac{5}{6}[/latex] | |
| [latex]\dfrac{12}{12}=\dfrac{6}{6}=\dfrac{3}{3}[/latex] | |
| [latex]\dfrac{1}{20}[/latex] | |
| [latex]\dfrac{2}{20}=\dfrac{1}{10}[/latex] | |
| [latex]\dfrac{2}{10}=\dfrac{1}{5}[/latex] | |
| [latex]\dfrac{4}{10}=\dfrac{2}{5}[/latex] | |
| [latex]\dfrac{6}{10}=\dfrac{3}{5}[/latex] | |
| [latex]\dfrac{8}{10}=\dfrac{4}{5}[/latex] | |
| [latex]\dfrac{10}{10}=\dfrac{5}{5}=1[/latex] |
Answers to Exercise 4
| Common Fraction | Decimal |
|---|---|
| [latex]\dfrac{1}{8}[/latex] | .0125 |
| [latex]\dfrac{2}{8}=\dfrac{1}{4}[/latex] | 0.25 |
| [latex]\dfrac{3}{8}[/latex] | 0.375 |
| [latex]\dfrac{4}{8}=\dfrac{2}{4}=\dfrac{1}{2}[/latex] | 0.5 |
| [latex]\dfrac{5}{8}[/latex] | 0.625 |
| [latex]\dfrac{6}{8}=\dfrac{3}{4}[/latex] | 0.75 |
| [latex]\dfrac{7}{8}[/latex] | 0.875 |
| [latex]\dfrac{8}{8}=\dfrac{4}{4}=\dfrac{2}{2}=1[/latex] |
1.0 |
| [latex]\dfrac{1}{12}[/latex] | [latex]0.08\overline{3}[/latex] |
| [latex]\dfrac{2}{12}=\dfrac{1}{6}[/latex] | [latex]0.1\overline{6}[/latex] |
| [latex]\dfrac{4}{12}=\dfrac{2}{6}=\dfrac{1}{3}[/latex] | [latex]0.\overline{3}[/latex] |
| [latex]\dfrac{6}{12}=\dfrac{3}{6}=\dfrac{1}{2}[/latex] | 0.5 |
| [latex]\dfrac{8}{12}=\dfrac{4}{6}=\dfrac{2}{3}[/latex] | [latex]0.\overline{6}[/latex] |
| [latex]\dfrac{10}{12}=\dfrac{5}{6}[/latex] | [latex]0.8\overline{3}[/latex] |
| [latex]\dfrac{12}{12}=\dfrac{6}{6}=\dfrac{3}{3}[/latex] | 1.0 |
| [latex]\dfrac{1}{20}[/latex] | 0.05 |
| [latex]\dfrac{2}{20}=\dfrac{1}{10}[/latex] | 0.1 |
| [latex]\dfrac{2}{10}=\dfrac{1}{5}[/latex] | 0.2 |
| [latex]\dfrac{4}{10}=\dfrac{2}{5}[/latex] | 0.4 |
| [latex]\dfrac{6}{10}=\dfrac{3}{5}[/latex] | 0.6 |
| [latex]\dfrac{8}{10}=\dfrac{4}{5}[/latex] | 0.8 |
| [latex]\dfrac{10}{10}=\dfrac{5}{5}=1[/latex] | 1.0 |
You may work with problems and real-life situations that use one decimal and one common fraction. Rewrite the fractions so both are decimals or both are common fractions. Choose the form that will give the answer the way it should be written.
Example D
Ted worked [latex]3\tfrac{3}{4}[/latex] hours at $8.25 per hour. How much did he earn? (Round to the nearest cent.)
The answer will be money which should be written using decimals, so work in the decimal form.
Rewrite [latex]3\tfrac{3}{4}[/latex] hours as 3.75 hours.
Ted earned [latex]3.75 × $8.25 = $30.94[/latex]
Example E
Jane cycled 49.4 km in [latex]2\tfrac{1}{2}[/latex] hours. What was her average speed?
The answer will be in km/hr. Metric measurements are written with decimals, so work in decimals.
Rewrite [latex]2\tfrac{1}{2}[/latex] hours as 2.5 hours and solve the problem.
[latex]49.4 ÷ 2.5 = 19.76[/latex] km/hr.
Topic A: Self-Test
Mark /11 Aim 9/11
- Complete the chart (7 marks).
Chart of common fractions and decimals to complete Common Fraction Decimal [latex]\dfrac{1}{4}[/latex] 0.125 0.3 [latex]\dfrac{3}{4}[/latex] 0.875 [latex]\dfrac{3}{5}[/latex] [latex]\dfrac{6}{6}[/latex] - Answer the following word problems (4 marks).
- Joseph worked hours a day, 5 days a week. He gets paid $9.35 per hour. How much does he get paid a week?
- Giang ran a 42.195 km marathon in hours. What was her average speed rounded to two decimal places?
Answers to Self Test A.
-
Chart of common fractions and decimals completed Common Fraction Decimal [latex]\dfrac{1}{4}[/latex] 0.25 [latex]\dfrac{1}{8}[/latex] 0.125 [latex]\dfrac{1}{3}[/latex] 0.3 [latex]\dfrac{3}{4}[/latex] 0.75 [latex]\dfrac{7}{8}[/latex] 0.875 [latex]\dfrac{3}{5}[/latex] 0.6 [latex]\dfrac{6}{6}[/latex] 1 -
- $268.81
- 9.93 km/hr