Question

Question was provided via image upload.

Explanation:

Step1: Set variable for $0.\overline{1}$

Let $x = 0.\overline{1} = 0.1111...$

Step2: Multiply by 10 to shift decimal

$10x = 1.1111...$

Step3: Subtract to eliminate repeating decimals

$10x - x = 1.1111... - 0.1111...$
$9x = 1$
$x = \frac{1}{9}$

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Step1: Set variable for $2.\overline{1}$

Let $x = 2.\overline{1} = 2.1111...$

Step2: Multiply by 10 to shift decimal

$10x = 21.1111...$

Step3: Subtract to eliminate repeating decimals

$10x - x = 21.1111... - 2.1111...$
$9x = 19$
$x = \frac{19}{9}$

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Step1: Set variable for $0.1\overline{6}$

Let $x = 0.1\overline{6} = 0.1666...$

Step2: Multiply by 10 to isolate non-repeating part

$10x = 1.6666...$

Step3: Multiply by 10 to shift repeating decimal

$100x = 16.6666...$

Step4: Subtract to eliminate repeating decimals

$100x - 10x = 16.6666... - 1.6666...$
$90x = 15$
$x = \frac{15}{90} = \frac{1}{6}$

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Step1: Set variable for $1.\overline{6}$

Let $x = 1.\overline{6} = 1.6666...$

Step2: Multiply by 10 to shift decimal

$10x = 16.6666...$

Step3: Subtract to eliminate repeating decimals

$10x - x = 16.6666... - 1.6666...$
$9x = 15$
$x = \frac{15}{9} = \frac{5}{3}$

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Step1: Set variable for $1.\overline{91}$

Let $x = 1.\overline{91} = 1.919191...$

Step2: Multiply by 100 to shift decimal

$100x = 191.919191...$

Step3: Subtract to eliminate repeating decimals

$100x - x = 191.919191... - 1.919191...$
$99x = 190$
$x = \frac{190}{99}$

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Step1: Simplify $0.\overline{36}$

For two repeating digits, use 99 as denominator:
$0.\overline{36} = \frac{36}{99} = \frac{4}{11}$

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Step1: Set variable for $3.\overline{6}$

Let $x = 3.\overline{6} = 3.6666...$

Step2: Multiply by 10 to shift decimal

$10x = 36.6666...$

Step3: Subtract to eliminate repeating decimals

$10x - x = 36.6666... - 3.6666...$
$9x = 33$
$x = \frac{33}{9} = \frac{11}{3}$

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Step1: Verify pre-done problems against answer bank

1. $0.\overline{19} = \frac{19}{99}$ (matches answer bank)
2. $0.\overline{210} = \frac{210}{999} = \frac{70}{333}$ (matches answer bank)
3. $0.\overline{2} = \frac{2}{9}$ (matches answer bank)
4. $0.\overline{16} = \frac{16}{99}$ (matches answer bank)
5. $0.\overline{90} = \frac{90}{99} = \frac{10}{11}$ (matches answer bank)
6. $0.\overline{91} = \frac{91}{99}$ (matches answer bank)
7. $0.\overline{162} = \frac{162}{999} = \frac{6}{37}$ (matches answer bank)
8. $0.\overline{369} = \frac{369}{999} = \frac{41}{111}$ (matches answer bank)

Answer:

1. $\frac{19}{99}$
2. $\frac{70}{333}$
3. $\frac{2}{9}$
4. $\frac{1}{9}$
5. $\frac{19}{9}$
6. $\frac{1}{6}$
7. $\frac{16}{99}$
8. $\frac{5}{3}$
9. $\frac{190}{99}$
10. $\frac{10}{11}$
11. $\frac{91}{99}$
12. $\frac{6}{37}$
13. $\frac{41}{111}$
14. $\frac{4}{11}$
15. $\frac{11}{3}$