Use the formula for the sum of an infinite geometric series to find a rational number in the form [tex]$\frac{a}{b}$[/tex], where [tex]$a$[/tex] and [tex]$b$[/tex] are integers and [tex]$b \neq 0$[/tex], for the following repeating decimals.

a. [tex]$0.7 \overline{2}$[/tex]
b. [tex]$3.0 \overline{72}$[/tex]

a. The fractional notation for [tex]$0.7 \overline{2}$[/tex] is [tex]$\boxed{\phantom{answer}}$[/tex]. (Type an integer or a simplified fraction.)



Answer :

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]