Answer :
To express the repeating decimal \( 1.\overline{2} \) as a fraction, we can follow these steps:
1. Let \( x \) represent the repeating decimal:
[tex]\[ x = 1.2222\ldots \][/tex]
2. Multiply both sides of the equation by 10 to move the decimal point one place to the right:
[tex]\[ 10x = 12.2222\ldots \][/tex]
3. Subtract the original equation \( x = 1.2222\ldots \) from this new equation to eliminate the repeating part:
[tex]\[ 10x - x = 12.2222\ldots - 1.2222\ldots \][/tex]
4. Perform the subtraction:
[tex]\[ 9x = 11 \][/tex]
5. Solve for \( x \) by dividing both sides by 9:
[tex]\[ x = \frac{11}{9} \][/tex]
Therefore, \( 1.\overline{2} \) as a fraction is \(\frac{11}{9}\).
To verify, let's convert \(\frac{11}{9}\) back to a decimal:
[tex]\[ \frac{11}{9} = 1.2222\ldots \][/tex]
This confirms that our fraction [tex]\(\frac{11}{9}\)[/tex] is correct.
1. Let \( x \) represent the repeating decimal:
[tex]\[ x = 1.2222\ldots \][/tex]
2. Multiply both sides of the equation by 10 to move the decimal point one place to the right:
[tex]\[ 10x = 12.2222\ldots \][/tex]
3. Subtract the original equation \( x = 1.2222\ldots \) from this new equation to eliminate the repeating part:
[tex]\[ 10x - x = 12.2222\ldots - 1.2222\ldots \][/tex]
4. Perform the subtraction:
[tex]\[ 9x = 11 \][/tex]
5. Solve for \( x \) by dividing both sides by 9:
[tex]\[ x = \frac{11}{9} \][/tex]
Therefore, \( 1.\overline{2} \) as a fraction is \(\frac{11}{9}\).
To verify, let's convert \(\frac{11}{9}\) back to a decimal:
[tex]\[ \frac{11}{9} = 1.2222\ldots \][/tex]
This confirms that our fraction [tex]\(\frac{11}{9}\)[/tex] is correct.
