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Here you will learn about converting a repeating decimal to a fraction including how to define a repeating decimal.
Students will first learn about converting repeating decimals (often called recurring decimals) to fractions as part of the number system in 8 th grade.
Converting a repeating decimal to fraction is representing the repeating decimals as a fraction without changing its value.
A repeating decimal (recurring decimal) is a decimal number that has a digit (or group of digits) right of the decimal point that repeats forever. The part that repeats can also be shown by placing a line over the first to the last digit of the repeating pattern. Repeating decimals are also known as non-terminating decimals.
For example,
0.\overline{3}=0.333333\ldots 4.\overline{24}=4.242424\ldots 10.\overline{123}=10.123123\ldots 6.5\overline{8}=6.588888\ldotsare all repeating decimals.
To convert a repeating decimal to a fraction, you can form an equation. E.g.
x=1.\overline{3} \hspace{1cm} โKeep track by labeling equations โ , โก, โข, etc.
You need to be able to multiply both sides of this equation confidently by different powers of 10. E.g.
10x=1.\overline{3} \hspace{1cm} โก 100x=13.\overline{3} \hspace{0.65cm} โขHow does this relate to 8 th grade math?
Use this quiz to check your grade 4 to 6 studentsโ understanding of converting fractions, decimals and percents. 10+ questions with answers covering a range of 4th, 5th and 6th grade converting fractions, decimals and percents topics to identify areas of strength and support!
DOWNLOAD FREEUse this quiz to check your grade 4 to 6 studentsโ understanding of converting fractions, decimals and percents. 10+ questions with answers covering a range of 4th, 5th and 6th grade converting fractions, decimals and percents topics to identify areas of strength and support!
DOWNLOAD FREEIn order to convert from a repeating decimal to a fraction, you need to:
Convert 0.\overline{1} to a fraction.
2Multiply both sides of \bf{โ } by a power of \bf{10} to create \bf{โก}.
As 0.\overline{1} has one repeating digit, multiply by 10. Remember, because you are multiplying the whole of โ by 10 , you also need to multiply the variable x by 10.
\begin{aligned}& 0.\overline{1}=x \hspace{0.8cm} โ \\\\ &0.\overline{1}\times 10=x\times10 \\\\ &1.\overline{1}=10x \hspace{0.5cm} โก \end{aligned}3Subtract \bf{โก} from \bf{โ }.
Subtract the first equation from the second equation.
\begin{aligned}&1.\overline{1}=10x \hspace{1cm} โก \\\\ -&0.\overline{1}=x \hspace{1.35cm} โ \\\\ &1.0=9x \end{aligned}4Divide the value by the coefficient of \textbf{x}.
\begin{aligned}&1=9x \\\\ &\cfrac{1}{9}=x\end{aligned}As x is equal to 0.\overline{1} and \cfrac{1}{9}, it can be stated that 0.\overline{1} and \cfrac{1}{9} are equal to one another.
5Simplify the fraction.
This fraction cannot be simplified as the only factor that 1 and 9 share is 1.
6State your answer.
0.\overline{1}=\cfrac{1}{9}Convert 0.\overline{12} to a fraction.
Equate the repeating decimal to a variable, \textbf{x}, to create \bf{โ }.
Multiply both sides of \bf{โ } by a power of \bf{10} to create \bf{โก}.
As 0.\overline{12} has two repeating digits, multiply by 100. Remember, because you are multiplying the whole of โ by 100, you also need to multiply the variable x by 100.
\begin{aligned}& 0.\overline{12}=x \hspace{1.2cm} โ \\\\ &0.\overline{12}\times 100=x\times100 \\\\ &12.\overline{12}=100x \hspace{0.5cm} โก \end{aligned}
Subtract \bf{โก} from \bf{โ }.
Subtract the first equation from the second equation.
\begin{aligned}&12.\overline{12}=100x \hspace{1cm} โก \\\\ -&0.\overline{12}=x \hspace{1.7cm} โ \\\\ &12=99x \end{aligned}
Divide the value by the coefficient of \textbf{x}.
Simplify the fraction.
3 is the greatest common factor of both 12 and 99 so, divide the numerator and the denominator by 3.
\cfrac{12\div3}{99\div3}=\cfrac{4}{33}
State your answer.
Convert 0.0\overline{1} to a fraction.
Equate the repeating decimal to a variable, \textbf{x}, to create \bf{โ }.
Multiply both sides of \bf{โ } by a power of \bf{10} to create \bf{โก}.
As 0.0\overline{1} has one repeating digit, multiply by 10. Remember, because you are multiplying the whole of โ by 10 , you also need to multiply the variable x by 10.
\begin{aligned}& 0.0\overline{1}=x \hspace{0.8cm} โ \\\\ &0.0\overline{1}\times 10=x\times10 \\\\
&0.1\overline{1}=10x \hspace{0.5cm} โก \end{aligned}
Subtract \bf{โก} from \bf{โ }.
Subtract the first equation from the second equation.
\begin{aligned}&0.1\overline{1}=10x \hspace{1cm} โก \\\\ -&0.0\overline{1}=x \hspace{1.35cm} โ \\\\ &0.1=9x \end{aligned}
Divide the value by the coefficient of \textbf{x}.
Simplify the fraction.
Here, notice that there is a decimal as the numerator, so multiply the numerator and the denominator by 10 to remove the decimal.
\cfrac{0.1\times10}{9\times10}=\cfrac{1}{90}
You cannot simplify the fraction \cfrac{1}{90}.
State the answer.
Convert 0.\overline{23} to a fraction.
Equate the repeating decimal to a variable, \textbf{x}, to create \bf{โ }.
Multiply both sides of \bf{โ } by a power of \bf{10} to create \bf{โก}.
As 0.\overline{23} has two repeating digits, multiply by 100. Remember, because you are multiplying the whole of โ by 100 , you also need to multiply the variable x by 100.
\begin{aligned}& 0.\overline{23}=x \hspace{1.5cm} โ \\\\ &0.\overline{23}\times 100=x\times100 \\\\ &23.\overline{23}=100x \hspace{0.8cm} โก \end{aligned}
Subtract \bf{โก} from \bf{โ }.
Subtract the first equation from the second equation.
\begin{aligned}&23.\overline{23}=100x \hspace{1cm} โก \\\\ -&0.\overline{23}=x \hspace{1.7cm} โ \\\\ &23.00=99x \end{aligned}
Divide the value by the coefficient of \textbf{x}.
Simplify the fraction.
This fraction cannot be simplified as the only factor that 23 and 99 share is 1.
State the answer.
Convert 8.\overline{7} to a fraction.
Equate the repeating decimal to a variable, \textbf{x}, to create \bf{โ }.
Multiply both sides of \bf{โ } by a power of \bf{10} to create \bf{โก}.
As 8.\overline{7} has one repeating digit, you will multiply by 10. Remember, because you are multiplying the whole of โ by 10 , you also need to multiply the variable x by 10.
\begin{aligned}& 8.\overline{7}=x \hspace{1.5cm} โ \\\\ &8.\overline{7}\times 10=x\times10 \\\\ &87.\overline{7}=10x \hspace{1cm} โก \end{aligned}
Subtract \bf{โก} from \bf{โ }.
Subtract the first equation from the second equation.
\begin{aligned}&87.\overline{7}=10x \hspace{1cm} โก \\\\ -&8.\overline{7}=x \hspace{1.55cm} โ \\\\ &79=9x \end{aligned}
Divide the value by the coefficient of \textbf{x}.
Simplify the fraction.
This fraction cannot be simplified as the only factor that 79 and 9 share is 1.
State the answer.
Convert 4.0\overline{46} to a fraction.
Equate the repeating decimal to a variable, \textbf{x}, to create \bf{โ }.
Multiply both sides of \bf{โ } by a power of \bf{10} to create \bf{โก}.
As 4.0 \overline{46} has two repeating digits, you will multiply by 100. Remember, because you are multiplying the whole of โ by 100 , you also need to multiply the variable x by 100.
\begin{aligned}& 4.0\overline{46}=x \hspace{1.85cm} โ \\\\ &4.0\overline{46}\times 100=x\times100 \\\\ &404.6\overline{46}=100x \hspace{1cm} โก \end{aligned}
Subtract \bf{โก} from \bf{โ }.
Subtract the first equation from the second equation.
\begin{aligned}&404.6\overline{46}=100x \hspace{0.65cm} โก \\\\ -&4.0\overline{46}=x \hspace{1.55cm} โ \\\\ &400.6=99x \end{aligned}
Divide the value by the coefficient of \textbf{x}.
Simplify the fraction.
There is a decimal as the numerator, so you will multiply the numerator (and denominator) by 10.
\cfrac{400.6\times10}{99\times10}=\cfrac{4006}{990}
You can simplify the fraction by dividing the numerator and the denominator by their greatest common factor, 2.
\cfrac{4006\div2}{990\div2}=\cfrac{2003}{495}
State the answer.
1. What should you multiply 0 . \overline{2} by to help eliminate the repeating decimal?
If the decimal has one repeating digit, then multiply by 10.
0.\overline{2}\times10=2.\overline{2}ย
2. What should you multiply 0.\overline{14} by to help eliminate the repeating decimal?
If the decimal has two repeating digits, then multiply by 100.
0.\overline{14}\times100=14.\overline{14}
3. What should you multiply 0.0\overline{15} by to help eliminate the repeating decimal?
If the decimal has two repeating digits, then multiply by 100.
0.0\overline{15}\times100=1.5\overline{15}
4. What is 0.\overline{2} ย as a fraction?
Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of 10.
\begin{aligned}&0.\overline{2}=x \hspace{1cm} โ \\\\ &0.\overline{2}\times10=x\times10 \\\\ &2.\overline{2}=10x \hspace{0.65cm} โก \end{aligned}
Subtract โ from โก.
\begin{aligned}&2.\overline{2}=10x \hspace{1cm} โก \\\\ -&0.\overline{2}=x \hspace{1.35cm} โ \\\\ &2.0=9x \end{aligned}
Divide both sides of the equation by 9
\cfrac{2}{9}=x
5. What is 0.\overline{34} ย as a fraction?
Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of 10.
\begin{aligned}&0.\overline{34}=x \hspace{2.0cm} โ \\\\ &0.\overline{34}\times100=x\times100 \\\\ &34.\overline{34}=100x \hspace{1.35cm} โก \end{aligned}
Subtract โ from โก.
\begin{aligned}&34.\overline{34}=100x \hspace{1cm} โก \\\\ -&0.\overline{34}=x \hspace{1.7cm} โ \\\\ &34.00=99x \end{aligned}
Divide both sides of the equation by 99 .
\cfrac{34}{99}=x
This does not simplify further.
6. What is 6.\overline{7} as a fraction?
Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of 10.
\begin{aligned}&6.\overline{7}=x \hspace{1cm} โ \\\\ &6.\overline{7}\times10=x\times10 \\\\ &67.\overline{7}=10x \hspace{0.65cm} โก \end{aligned}
Subtract โ from โก.
\begin{aligned}&67.\overline{7}=10x \hspace{1cm} โก \\\\ -&6.\overline{7}=x \hspace{1.55cm} โ \\\\ &61.0=9x \end{aligned}
Divide both sides of the equation by 9 .
\cfrac{61}{9}=x
This does not simplify further.
No, both terms can be used synonymously.
A surd is an irrational number expressed in radical form. They cannot be simplified to whole numbers or fractions.
Terminating decimals are decimals that contain a finite number of digits after the decimal point.
Yes. These are called irrational numbers and include pi (\pi=3.14159265\ldots) and Eulerโs number (e=2.7182818284590\ldots).
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