Question

1. select all the numbers that are equivalent to $0.8\overline{3}$.$square$ $\frac{5}{6}LXB0square$ $\frac{83}{100}LXB1square$ $83.\overline{3}\\%$9. write $11.\overline{1}\\%$ as a fraction and as a decimal.10. which number is not equivalent to $2.\overline{3}$?$circled{a}$ $2\frac{1}{3}LXB2circled{c}$ $2.\overline{3}\\%$$circled{d}$ $233.\overline{3}\\%$11. write $\frac{5}{11}$ as a decimal and as a percent.

Explanation:

Question 8

Step1: Convert $0.\overline{83}$ to fraction

Let $x = 0.\overline{83} = 0.838383...$
Multiply by 100: $100x = 83.838383...$
Subtract: $100x - x = 83.8383... - 0.8383...$
$99x = 83 \implies x = \frac{83}{99} \approx 0.838383...$

Step2: Evaluate each option

• $\frac{5}{6} \approx 0.8333...$ (not equal)
• $\frac{10}{12} = \frac{5}{6} \approx 0.8333...$ (not equal)
• $\frac{83}{100} = 0.83$ (not equal)
• $8.\overline{3}\% = \frac{8.\overline{3}}{100} = 0.08333...$ (not equal)
• $83.\overline{3}\% = \frac{83.\overline{3}}{100} = 0.8333...$ (not equal)

*Correction: Recheck $0.\overline{83}$ conversion error:
Let $x=0.8\overline{3}$ (bar only on 3):
$10x=8.\overline{3}$, $100x=83.\overline{3}$
$100x-10x=75 \implies 90x=75 \implies x=\frac{75}{90}=\frac{5}{6} \approx 0.8333...$
Re-evaluate options:

• $\frac{5}{6} = 0.8333... = 0.8\overline{3}$ (equal)
• $\frac{10}{12} = \frac{5}{6}$ (equal)
• $\frac{83}{100}=0.83$ (not equal)
• $8.\overline{3}\%=0.08333...$ (not equal)
• $83.\overline{3}\% = 0.8333... = 0.8\overline{3}$ (equal)

Step1: Convert $11.\overline{1}\%$ to decimal

$11.\overline{1}\% = \frac{11.\overline{1}}{100}$
Let $x=11.\overline{1}=11.111...$, $10x=111.111...$
$10x-x=100 \implies 9x=100 \implies x=\frac{100}{9}$
Decimal: $\frac{100}{9} \div 100 = \frac{1}{9} \approx 0.111...$

Step2: Convert to fraction

$11.\overline{1}\% = \frac{11.\overline{1}}{100} = \frac{\frac{100}{9}}{100} = \frac{1}{9}$

Step1: Convert $2.\overline{3}$ to fraction/percent

Let $x=2.\overline{3}=2.333...$, $10x=23.333...$
$10x-x=21 \implies 9x=21 \implies x=\frac{21}{9}=\frac{7}{3}=2\frac{1}{3}$
Percent: $2.\overline{3} \times 100 = 233.\overline{3}\%$

Step2: Evaluate each option

• A: $2\frac{1}{3} = \frac{7}{3} = 2.\overline{3}$ (equal)
• B: $\frac{7}{3} = 2.\overline{3}$ (equal)
• C: $2.\overline{3}\% = 0.02333...$ (not equal)
• D: $233.\overline{3}\% = 2.\overline{3}$ (equal)

Answer:

$\frac{5}{6}$, $\frac{10}{12}$, $83.\overline{3}\%$

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Question 9