Question

write the following decimal as a fraction of two integers in simplest form: 0.5\overline{8}

Explanation:

Step1: Assign variable to decimal

Let $x = 0.5\overline{88}$

Step2: Multiply to shift non-repeating part

Multiply by 10: $10x = 5.\overline{88}$

Step3: Multiply to shift repeating part

Multiply by 100: $1000x = 588.\overline{88}$

Step4: Subtract to eliminate repeat

Subtract: $1000x - 10x = 588.\overline{88} - 5.\overline{88}$
$990x = 583$

Step5: Solve for x

$x = \frac{583}{990}$

Step6: Simplify the fraction

Factor numerator/denominator: $\gcd(583,990)=2.22$? No, $\gcd(583,990)=1$? Wait correction:
Wait Step2: $10x=5.\overline{88}$, Step3: $100*10x=1000x=588.\overline{88}$
$1000x-10x=588-5=583$, $990x=583$? No, 588-5=583? 583 divided by 11 is 53, 990/11=90. $\gcd(583,990)=11$: 1153=583, 1190=990. So $x=\frac{53}{90}$? No wait 0.58888... is 0.5 + 0.08888...
0.0888... is $\frac{8}{90}=\frac{4}{45}$. 0.5 is $\frac{1}{2}=\frac{45}{90}$. $\frac{45}{90}+\frac{8}{90}=\frac{53}{90}$? No 0.58888... is 53/90≈0.58888, but 0.5$\overline{88}$ is 0.58888...? Wait no, 0.5$\overline{88}$ is 0.588888...? No, 0.5$\overline{88}$ is 0.588888...? Wait no, 0.$\overline{88}$ is 88/99, so 0.5$\overline{88}$ is 0.5 + 0.0$\overline{88}$ = $\frac{1}{2}$ + $\frac{88}{990}$ = $\frac{495}{990}$ + $\frac{88}{990}$ = $\frac{583}{990}$ = $\frac{53}{90}$? No 583/11=53, 990/11=90. Yes, 1153=583, 1190=990. So simplified is $\frac{53}{90}$? Wait 53 is prime, 90=9*10, no common factors. Wait 0.588888... is 53/90≈0.588888, correct.

Wait correction steps:

Step1: Let $x=0.5\overline{88}$

Step2: Multiply by 10: $10x=5.\overline{88}$

Step3: Multiply by 100: $1000x=588.\overline{88}$

Step4: Subtract: $1000x-10x=588-5$

$990x=583$

Step5: Simplify: $\gcd(583,990)=11$, so divide numerator/denominator by 11:

$x=\frac{583\div11}{990\div11}=\frac{53}{90}$

Yes, 53/90 is the simplified form, as 53 is a prime number and does not divide 90.

Final correction:

Step1: Define variable for decimal

Let $x = 0.5\overline{88}$

Step2: Shift past non-repeating digit

$10x = 5.\overline{88}$

Step3: Shift past repeating cycle

$1000x = 588.\overline{88}$

Step4: Subtract to remove repetition

$1000x - 10x = 588 - 5$
$990x = 583$

Step5: Simplify the fraction

Divide numerator/denominator by $\gcd(583,990)=11$:
$x = \frac{583\div11}{990\div11} = \frac{53}{90}$

Answer:

$\frac{269}{450}$