Question

write (0.overline{16}) as a fraction.
a (\frac{1}{12})
b (\frac{1}{3})
c (\frac{1}{6})
d (\frac{1}{9})

Explanation:

Step1: Assign variable to repeating decimal

Let $x = 0.\overline{16}$, which means $x = 0.161616...$

Step2: Eliminate repeating decimals

Multiply $x$ by 100 (since 2 digits repeat):
$100x = 16.161616...$

Step3: Subtract to isolate $x$

Subtract original $x$ from $100x$:
$100x - x = 16.161616... - 0.161616...$
$99x = 16$

Step4: Solve for $x$

$x = \frac{16}{99}$
Wait, correct the options check: Wait, no—wait, if it's $0.\overline{6}$ it's $\frac{2}{3}$, but $0.\overline{16}$ is $\frac{16}{99}$, which is not listed. Wait, typo? Wait, no—wait, maybe it's $0.1\overline{6}$? Let's check:
If $x=0.1\overline{6}=0.1666...$
Multiply by 10: $10x=1.666...$
Multiply by 100: $100x=16.666...$
Subtract: $100x-10x=16.666...-1.666...$
$90x=15$
$x=\frac{15}{90}=\frac{1}{6}$

Wait, the question says $0.\overline{16}$, but option C is $\frac{1}{6}$. Wait, maybe the bar is only over 6? Assuming the question has a formatting error and it's $0.1\overline{6}$ (common problem), then:

Step1: Define $x$ for $0.1\overline{6}$

Let $x = 0.1666...$

Step2: Multiply by 10

$10x = 1.666...$

Step3: Multiply by 100

$100x = 16.666...$

Step4: Subtract to eliminate repeat

$100x - 10x = 16.666... - 1.666...$
$90x = 15$

Step5: Simplify fraction

$x = \frac{15}{90} = \frac{1}{6}$

Answer:

C. $\frac{1}{6}$