To express the repeating decimal [tex]\(1 . \overline{48}\)[/tex] as a mixed number, follow these steps:
1. Express the repeating decimal with a variable:
Let [tex]\( x = 1.484848\ldots \)[/tex]
2. Eliminate the repeating part:
Multiply [tex]\( x \)[/tex] by 100 (since the repeating block "48" contains two digits):
[tex]\[
100x = 148.484848\ldots
\][/tex]
3. Formulate the equation by subtracting:
Subtract the original [tex]\( x \)[/tex] from the equation above to eliminate the repeating decimal part:
[tex]\[
100x - x = 148.484848\ldots - 1.484848\ldots
\][/tex]
This simplifies to:
[tex]\[
99x = 147
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by 99 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{147}{99}
\][/tex]
5. Simplify the fraction:
Determine the greatest common divisor (GCD) of 147 and 99 to simplify the fraction:
The GCD of 147 and 99 is 3.
Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{147 \div 3}{99 \div 3} = \frac{49}{33}
\][/tex]
6. Express the result as a mixed number:
Since [tex]\( x \)[/tex] initially represented [tex]\( 1.484848\ldots \)[/tex], and we separated it into an integer part and a fraction, we add the integer part back to the simplified fraction:
[tex]\[
x = 1 + \frac{49}{33}
\][/tex]
Therefore, [tex]\( 1 . \overline{48} \)[/tex] as a mixed number is:
[tex]\[
1 \frac{49}{33}
\][/tex]
