Answer:
[tex]4.\overline{2}\longleftrightarrow 4\frac29[/tex]
[tex]0.\overline{79}\longleftrightarrow \dfrac{79}{99}[/tex]
[tex]-0.3\overline{6}\longleftrightarrow -\dfrac{11}{30}[/tex]
[tex]-0.5\overline{2}\longleftrightarrow -\dfrac{47}{90}[/tex]
[tex]0.\overline{7}\longleftrightarrow \dfrac79[/tex]
[tex]4.\overline{21}\longleftrightarrow 4\frac{7}{33}[/tex]
Step-by-step explanation:
A repeating decimal is a decimal number with a digit (or group of digits) that repeats forever.
To express a repeating decimal as a rational number:
- Create equation 1 by letting x equal the repeating decimal.
- Create equation 2 by multiplying both sides of the first equation by 10ⁿ, where n is the number of digits in the repeating part of the repeating decimal. This shifts the repeating part after the decimal point.
- If there are non-repeating digits after the decimal point, create equation 3 by multiplying equation 2 by 10ⁿ, where n is the number of digits in the repeating part. This ensures that all non-repeating digits are shifted after the decimal point.
- Subtract the first equation from the second equation (or equation 2 from equation 3, where applicable) to eliminate the part after the decimal.
- Rearrange the resulting equation and solve for x.
- If possible, reduce the fraction to is simplest form by dividing the numerator and denominator by their greatest common factor.
[tex]\dotfill[/tex]
[tex]\LARGE\text{$\boxed{4.\overline{2}}$}[/tex]
Here, the repeating digit is 2. As there is one digit in the repeating part, we multiply by 10¹ = 10.
[tex]\textsf{Equation 1:}\quad x=4.\overline{2}\\\\\textsf{Equation 2:}\quad 10x=42.\overline{2}[/tex]
Subtract the first equation from the second equation:
[tex]10x-x=42.\overline{2}-4.\overline{2}\\\\9x=38[/tex]
Solve for x:
[tex]x=\dfrac{38}{9}[/tex]
Express as a mixed number:
[tex]x=\dfrac{36+2}{9}=\dfrac{36}{9}+\dfrac{2}{9}=4\frac29[/tex]
Therefore:
[tex]\LARGE\text{$4.\overline{2}\longleftrightarrow 4\frac29$}[/tex]
[tex]\dotfill[/tex]
[tex]\LARGE\text{$\boxed{0.\overline{79}}$}[/tex]
Here, the repeating digits are 79. As there is two digits in the repeating part, we multiply by 10² = 100.
[tex]\textsf{Equation 1:}\quad x=0.\overline{79}\\\\\textsf{Equation 2:}\quad 100x=79.\overline{79}[/tex]
Subtract the first equation from the second equation:
[tex]100x-x=79.\overline{79}-0.\overline{79}\\\\99x=79[/tex]
Solve for x:
[tex]x=\dfrac{79}{99}[/tex]
Therefore:
[tex]\LARGE\text{$0.\overline{79}\longleftrightarrow \dfrac{79}{99}$}[/tex]
[tex]\dotfill[/tex]
[tex]\LARGE\text{$\boxed{-0.3\overline{6}}$}[/tex]
Here, the repeating digit is 6. As there is also one non-repeating digit after the decimal point, we need to create the three equations:
[tex]\textsf{Equation 1:}\quad x=-0.3\overline{6}\\\\\textsf{Equation 2:}\quad 10x=-3.\overline{6}\\\\\textsf{Equation 3:}\quad 100x=-36.\overline{6}[/tex]
Subtract the second equation from the third equation:
[tex]100x-10x=-36.\overline{6}-(-3.\overline{6})\\\\90x=-33[/tex]
Solve for x:
[tex]x=-\dfrac{33}{90}[/tex]
Reduce the fraction to is simplest form:
[tex]x=-\dfrac{33 \div 3}{90\div 3}=-\dfrac{11}{30}[/tex]
Therefore:
[tex]\LARGE\text{$-0.3\overline{6}\longleftrightarrow -\dfrac{11}{30}$}[/tex]
[tex]\dotfill[/tex]
[tex]\LARGE\text{$\boxed{-0.5\overline{2}}$}[/tex]
Here, the repeating digit is 2. As there is also one non-repeating digit after the decimal point, we need to create the three equations:
[tex]\textsf{Equation 1:}\quad x=-0.5\overline{2}\\\\\textsf{Equation 2:}\quad 10x=-5.\overline{2}\\\\\textsf{Equation 3:}\quad 100x=-52.\overline{2}[/tex]
Subtract the second equation from the third equation:
[tex]100x-10x=-52.\overline{2}-(-5.\overline{2})\\\\90x=-47[/tex]
Solve for x:
[tex]x=-\dfrac{47}{90}[/tex]
Therefore:
[tex]\LARGE\text{$-0.5\overline{2}\longleftrightarrow -\dfrac{47}{90}$}[/tex]
[tex]\dotfill[/tex]
[tex]\LARGE\text{$\boxed{0.\overline{7}}$}[/tex]
Here, the repeating digit is 7. As there is one digit in the repeating part, we multiply by 10¹ = 10.
[tex]\textsf{Equation 1:}\quad x=0.\overline{7}\\\\\textsf{Equation 2:}\quad 10x=7.\overline{7}[/tex]
Subtract the first equation from the second equation:
[tex]10x-x=7.\overline{7}-0.\overline{7}\\\\9x=7[/tex]
Solve for x:
[tex]x=\dfrac{7}{9}[/tex]
Therefore:
[tex]\LARGE\text{$0.\overline{7}\longleftrightarrow \dfrac79$}[/tex]
[tex]\dotfill[/tex]
[tex]\LARGE\text{$\boxed{4.\overline{21}}$}[/tex]
Here, the repeating digits are 21. As there is two digits in the repeating part, we multiply by 10² = 100.
[tex]\textsf{Equation 1:}\quad x=4.\overline{21}\\\\\textsf{Equation 2:}\quad 100x=421.\overline{21}[/tex]
Subtract the first equation from the second equation:
[tex]100x-x=421.\overline{21}-4.\overline{21}\\\\99x=417[/tex]
Solve for x:
[tex]x=\dfrac{417}{99}[/tex]
Reduce the fraction to is simplest form:
[tex]x=\dfrac{417\div 3}{99\div 3}=\dfrac{139}{33}[/tex]
Express as a mixed number:
[tex]x=\dfrac{132+7}{33}=\dfrac{132}{33}+\dfrac{7}{33}=4\frac{7}{33}[/tex]
Therefore:
[tex]\LARGE\text{$4.\overline{21}\longleftrightarrow 4\frac{7}{33}$}[/tex]
