To convert the repeating decimal [tex]\( 1.\overline{45} \)[/tex] into a fraction, follow these steps:
1. Express the repeating decimal as a variable:
Let [tex]\( x = 1.\overline{45} \)[/tex], which means:
[tex]\[
x = 1.45454545\ldots
\][/tex]
2. Multiply by a power of 10 to shift the decimal point:
Since the decimal sequence "45" repeats every 2 digits, multiply [tex]\( x \)[/tex] by [tex]\( 10^2 = 100 \)[/tex]:
[tex]\[
100x = 145.45454545\ldots
\][/tex]
3. Set up a subtraction to eliminate the repeating part:
Subtract the original variable [tex]\( x \)[/tex] from this equation:
[tex]\[
100x - x = 145.45454545\ldots - 1.45454545\ldots
\][/tex]
This simplifies to:
[tex]\[
99x = 144
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{144}{99}
\][/tex]
5. Simplify the fraction:
Find the greatest common divisor (GCD) of 144 and 99 to reduce the fraction to its simplest form. The GCD of 144 and 99 is 9:
[tex]\[
\frac{144}{99} = \frac{144 \div 9}{99 \div 9} = \frac{16}{11}
\][/tex]
6. Verify the result:
The simplified fraction is [tex]\( \frac{16}{11} \)[/tex].
So, the repeating decimal [tex]\( 1.\overline{45} \)[/tex] can be written as the fraction:
[tex]\[
\frac{16}{11}
\][/tex]
