Question

1. write $\frac{1}{13}$ as a decimal and as a percent
2. what is the difference between $0.1\overline{5}$ and $0.\overline{15}$? write the fraction equivalent for each number.
3. what is $0.\overline{8}$ as a percent?

Explanation:

Step1: Convert $\frac{1}{15}$ to decimal

$1 \div 15 = 0.0\overline{6}$

Step2: Convert decimal to percent

$0.0\overline{6} \times 100 = 6.\overline{6}\%$

Step3: Analyze $0.1\overline{5}$ (repeating 5)

Let $x = 0.1\overline{5}$, so $10x = 1.\overline{5}$ and $100x = 15.\overline{5}$. Subtract: $100x - 10x = 15.\overline{5} - 1.\overline{5}$ → $90x = 14$ → $x = \frac{14}{90} = \frac{7}{45}$

Step4: Analyze $0.\overline{15}$ (repeating 15)

Let $y = 0.\overline{15}$, so $100y = 15.\overline{15}$. Subtract: $100y - y = 15.\overline{15} - 0.\overline{15}$ → $99y = 15$ → $y = \frac{15}{99} = \frac{5}{33}$

Step5: State the difference

$0.1\overline{5}$ has only the 5 repeating (after the tenths place), while $0.\overline{15}$ has both 1 and 5 repeating.

Step6: Convert $0.\overline{8}$ to fraction

Let $z = 0.\overline{8}$, so $10z = 8.\overline{8}$. Subtract: $10z - z = 8.\overline{8} - 0.\overline{8}$ → $9z = 8$ → $z = \frac{8}{9}$

Step7: Convert fraction to percent

$\frac{8}{9} \times 100 = 88.\overline{8}\%$

Answer:

1. Decimal: $0.0\overline{6}$, Percent: $6.\overline{6}\%$
2. Difference: $0.1\overline{5}$ is a decimal where only the digit 5 repeats (after the 1 in the tenths place), while $0.\overline{15}$ is a decimal where the two-digit sequence "15" repeats infinitely.

• Fraction for $0.1\overline{5}$: $\frac{7}{45}$
• Fraction for $0.\overline{15}$: $\frac{5}{33}$

1. $88.\overline{8}\%$