Sure, let's break down the process for converting the repeating decimal [tex]\(1.\overline{48}\)[/tex] into a mixed number step-by-step.
1. Identify the components of the number:
- Integer part: 1
- Repeating decimal part: [tex]\(0.\overline{48}\)[/tex]
2. Express the repeating decimal as a fraction:
- Let [tex]\( x = 0.\overline{48} \)[/tex].
- To eliminate the repeating part, multiply [tex]\( x \)[/tex] by 100 (since the repeating part has two digits):
[tex]\[
100x = 48.\overline{48}
\][/tex]
- Now, we have two equations:
[tex]\[
x = 0.\overline{48}
\][/tex]
[tex]\[
100x = 48.\overline{48}
\][/tex]
- Subtract the first equation from the second:
[tex]\[
100x - x = 48.\overline{48} - 0.\overline{48}
\][/tex]
[tex]\[
99x = 48
\][/tex]
- Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = \frac{48}{99}
\][/tex]
- Simplify the fraction by finding the greatest common divisor (GCD) of 48 and 99, which is 3:
[tex]\[
\frac{48 \div 3}{99 \div 3} = \frac{16}{33}
\][/tex]
- Therefore, [tex]\(0.\overline{48} = \frac{16}{33}\)[/tex].
3. Combine the integer part and the fraction part:
- The integer part is 1.
- The fraction part is [tex]\(\frac{16}{33}\)[/tex].
4. Form the mixed number:
- Combine the two parts to get:
[tex]\[
1 \frac{16}{33}
\][/tex]
Thus, the repeating decimal [tex]\(1.\overline{48}\)[/tex] expressed as a mixed number is:
[tex]\[
1 \frac{16}{33}
\][/tex]
