Recurring decimals questions and answers

Worked solution

1. $x = 0.444\ldots$ and $10x = 4.444\ldots$
2. Subtract: $9x = 4$
3. $x = \tfrac{4}{9}$

Worked solution

1. $x = 0.2727\ldots$ and $100x = 27.2727\ldots$
2. Subtract: $99x = 27$, so $x = \tfrac{27}{99}$
3. Simplify (divide by 9): $\tfrac{3}{11}$

Worked solution

1. $x = 1.333\ldots$ and $10x = 13.333\ldots$
2. Subtract: $9x = 12$, so $x = \tfrac{12}{9} = \tfrac{4}{3}$

Worked solution

1. $x = 0.1666\ldots$, $10x = 1.666\ldots$, $100x = 16.666\ldots$
2. Subtract: $90x = 15$, so $x = \tfrac{15}{90} = \tfrac{1}{6}$

Worked solution

1. $10x = 4.555\ldots$ and $100x = 45.555\ldots$
2. Subtract: $90x = 41$, so $x = \tfrac{41}{90}$

Worked solution

1. $10x = 0.777\ldots$ and $100x = 7.777\ldots$
2. Subtract: $90x = 7$, so $x = \tfrac{7}{90}$

Worked solution

1. $x = 0.123123\ldots$ and $1000x = 123.123123\ldots$
2. Subtract: $999x = 123$, so $x = \tfrac{123}{999}$
3. Divide by 3: $\tfrac{41}{333}$

Worked solution

1. $5 \div 11 = 0.454545\ldots$
2. The digits 4 and 5 both repeat, so it's $0.\dot{4}\dot{5}$.

Worked solution

1. $x = 0.23636\ldots$, $10x = 2.3636\ldots$, $1000x = 236.3636\ldots$
2. Subtract: $990x = 234$, so $x = \tfrac{234}{990}$
3. Divide by 18: $\tfrac{13}{55}$

Worked solution

1. Let $x = 0.999\ldots$
2. Then $10x = 9.999\ldots$
3. Subtract: $9x = 9$
4. So $x = 1$, which means $0.\dot{9} = 1$.