To convert the repeating decimals into fractions, we can use algebraic methods that are essentially based on the formula for the sum of an infinite geometric series.
### Problem (a): [tex]\(0.7 \overline{2} \)[/tex]
Let's denote the repeating decimal [tex]\(0.7 \overline{2} \)[/tex] as [tex]\( x \)[/tex].
1. Write the equation for [tex]\( x \)[/tex]:
[tex]\[
x = 0.722222\ldots
\][/tex]
2. To remove the repeating part, multiply both sides of the equation by 10:
[tex]\[
10x = 7.22222\ldots
\][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[
10x - x = 7.22222\ldots - 0.72222\ldots
\][/tex]
4. Simplify the equation:
[tex]\[
9x = 6.5
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{6.5}{9}
\][/tex]
Simplifying [tex]\( \frac{6.5}{9} \)[/tex] further, we note:
[tex]\[
6.5 = \frac{13}{2}
\][/tex]
Thus:
[tex]\[
x = \frac{\frac{13}{2}}{9} = \frac{13}{2} \cdot \frac{1}{9} = \frac{13}{18}
\][/tex]
So, the fractional notation for [tex]\(0.7 \overline{2} \)[/tex] is:
[tex]\[
\boxed{\frac{13}{18}}
\][/tex]
### Problem (b): [tex]\(3.0 \overline{72} \)[/tex]
Let's denote the repeating decimal [tex]\(3.0 \overline{72} \)[/tex] as [tex]\( y \)[/tex].
1. Write the equation for [tex]\( y \)[/tex]:
[tex]\[
y = 3.0727272\ldots
\][/tex]
2. To remove the repeating part, multiply both sides of the equation by 1000 to include the three repeating digits:
[tex]\[
1000y = 3072.7272\ldots
\][/tex]
3. Also, multiply [tex]\( y \)[/tex] by 10 to handle the non-repeating part preceding the repeating part:
[tex]\[
10y = 30.7272\ldots
\][/tex]
4. Subtract the equation for [tex]\( 10y \)[/tex] from the equation for [tex]\( 1000y \)[/tex]:
[tex]\[
1000y - 10y = 3072.7272\ldots - 30.7272\ldots
\][/tex]
5. Simplify the equation:
[tex]\[
990y = 3042
\][/tex]
6. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3042}{990}
\][/tex]
7. Simplify [tex]\( \frac{3042}{990} \)[/tex]:
The greatest common divisor (GCD) of 3042 and 990 is 6. Therefore:
[tex]\[
y = \frac{3042 \div 6}{990 \div 6} = \frac{507}{165} = \frac{169}{55}
\][/tex]
So, the fractional notation for [tex]\(3.0 \overline{72} \)[/tex] is:
[tex]\[
\boxed{\frac{169}{55}}
\][/tex]
