Convert Repeating Decimals to Fractions (examples, solutions ... - Free Printable

To convert repeating decimals to fractions, we use algebraic methods. Let's solve each problem step by step.

1. Convert \(0.\overline{3}\) to a fraction:

Let \( x = 0.\overline{3} \).
This means \( x = 0.3333\ldots \).

Multiply both sides by 10:
\[ 10x = 3.3333\ldots \]

Now subtract the original equation \( x = 0.3333\ldots \) from this:
\[ 10x - x = 3.3333\ldots - 0.3333\ldots \]
\[ 9x = 3 \]
\[ x = \frac{3}{9} = \frac{1}{3} \]

So, \( 0.\overline{3} = \frac{1}{3} \).

2. Convert \(0.\overline{6}\) to a fraction:

Let \( x = 0.\overline{6} \).
This means \( x = 0.6666\ldots \).

Multiply both sides by 10:
\[ 10x = 6.6666\ldots \]

Now subtract the original equation \( x = 0.6666\ldots \) from this:
\[ 10x - x = 6.6666\ldots - 0.6666\ldots \]
\[ 9x = 6 \]
\[ x = \frac{6}{9} = \frac{2}{3} \]

So, \( 0.\overline{6} = \frac{2}{3} \).

3. Convert \(0.\overline{1}\) to a fraction:

Let \( x = 0.\overline{1} \).
This means \( x = 0.1111\ldots \).

Multiply both sides by 10:
\[ 10x = 1.1111\ldots \]

Now subtract the original equation \( x = 0.1111\ldots \) from this:
\[ 10x - x = 1.1111\ldots - 0.1111\ldots \]
\[ 9x = 1 \]
\[ x = \frac{1}{9} \]

So, \( 0.\overline{1} = \frac{1}{9} \).

4. Convert \(0.\overline{2}\) to a fraction:

Let \( x = 0.\overline{2} \).
This means \( x = 0.2222\ldots \).

Multiply both sides by 10:
\[ 10x = 2.2222\ldots \]

Now subtract the original equation \( x = 0.2222\ldots \) from this:
\[ 10x - x = 2.2222\ldots - 0.2222\ldots \]
\[ 9x = 2 \]
\[ x = \frac{2}{9} \]

So, \( 0.\overline{2} = \frac{2}{9} \).

5. Convert \(0.\overline{32}\) to a fraction:

Let \( x = 0.\overline{32} \).
This means \( x = 0.323232\ldots \).

Multiply both sides by 100 (since the repeating block has 2 digits):
\[ 100x = 32.323232\ldots \]

Now subtract the original equation \( x = 0.323232\ldots \) from this:
\[ 100x - x = 32.323232\ldots - 0.323232\ldots \]
\[ 99x = 32 \]
\[ x = \frac{32}{99} \]

So, \( 0.\overline{32} = \frac{32}{99} \).

6. Convert \(0.\overline{42}\) to a fraction:

Let \( x = 0.\overline{42} \).
This means \( x = 0.424242\ldots \).

Multiply both sides by 100 (since the repeating block has 2 digits):
\[ 100x = 42.424242\ldots \]

Now subtract the original equation \( x = 0.424242\ldots \) from this:
\[ 100x - x = 42.424242\ldots - 0.424242\ldots \]
\[ 99x = 42 \]
\[ x = \frac{42}{99} = \frac{14}{33} \]

So, \( 0.\overline{42} = \frac{14}{33} \).

7. Convert \(0.1\overline{6}\) to a fraction:

Let \( x = 0.1\overline{6} \).
This means \( x = 0.16666\ldots \).

First, let's handle the non-repeating part. Write \( x \) as:
\[ x = 0.1 + 0.0\overline{6} \]

We already know that \( 0.\overline{6} = \frac{2}{3} \), so:
\[ 0.0\overline{6} = \frac{2}{3} \times \frac{1}{10} = \frac{2}{30} = \frac{1}{15} \]

Now add the non-repeating part:
\[ x = 0.1 + \frac{1}{15} = \frac{1}{10} + \frac{1}{15} \]

Find a common denominator (30):
\[ x = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \]

So, \( 0.1\overline{6} = \frac{1}{6} \).

8. Convert \(0.8\overline{3}\) to a fraction:

Let \( x = 0.8\overline{3} \).
This means \( x = 0.83333\ldots \).

First, let's handle the non-repeating part. Write \( x \) as:
\[ x = 0.8 + 0.0\overline{3} \]

We already know that \( 0.\overline{3} = \frac{1}{3} \), so:
\[ 0.0\overline{3} = \frac{1}{3} \times \frac{1}{10} = \frac{1}{30} \]

Now add the non-repeating part:
\[ x = 0.8 + \frac{1}{30} = \frac{8}{10} + \frac{1}{30} = \frac{4}{5} + \frac{1}{30} \]

Find a common denominator (30):
\[ x = \frac{24}{30} + \frac{1}{30} = \frac{25}{30} = \frac{5}{6} \]

So, \( 0.8\overline{3} = \frac{5}{6} \).

9. Convert \(0.7\overline{6}\) to a fraction:

Let \( x = 0.7\overline{6} \).
This means \( x = 0.76666\ldots \).

First, let's handle the non-repeating part. Write \( x \) as:
\[ x = 0.7 + 0.0\overline{6} \]

We already know that \( 0.\overline{6} = \frac{2}{3} \), so:
\[ 0.0\overline{6} = \frac{2}{3} \times \frac{1}{10} = \frac{2}{30} = \frac{1}{15} \]

Now add the non-repeating part:
\[ x = 0.7 + \frac{1}{15} = \frac{7}{10} + \frac{1}{15} \]

Find a common denominator (30):
\[ x = \frac{21}{30} + \frac{2}{30} = \frac{23}{30} \]

So, \( 0.7\overline{6} = \frac{23}{30} \).

10. Convert \(0.6\overline{5}\) to a fraction:

Let \( x = 0.6\overline{5} \).
This means \( x = 0.65555\ldots \).

First, let's handle the non-repeating part. Write \( x \) as:
\[ x = 0.6 + 0.0\overline{5} \]

We already know that \( 0.\overline{5} = \frac{5}{9} \), so:
\[ 0.0\overline{5} = \frac{5}{9} \times \frac{1}{10} = \frac{5}{90} = \frac{1}{18} \]

Now add the non-repeating part:
\[ x = 0.6 + \frac{1}{18} = \frac{6}{10} + \frac{1}{18} = \frac{3}{5} + \frac{1}{18} \]

Find a common denominator (90):
\[ x = \frac{54}{90} + \frac{5}{90} = \frac{59}{90} \]

So, \( 0.6\overline{5} = \frac{59}{90} \).

11. Convert \(0.31\overline{6}\) to a fraction:

Let \( x = 0.31\overline{6} \).
This means \( x = 0.316666\ldots \).

First, let's handle the non-repeating part. Write \( x \) as:
\[ x = 0.31 + 0.00\overline{6} \]

We already know that \( 0.\overline{6} = \frac{2}{3} \), so:
\[ 0.00\overline{6} = \frac{2}{3} \times \frac{1}{100} = \frac{2}{300} = \frac{1}{150} \]

Now add the non-repeating part:
\[ x = 0.31 + \frac{1}{150} = \frac{31}{100} + \frac{1}{150} \]

Find a common denominator (300):
\[ x = \frac{93}{300} + \frac{2}{300} = \frac{95}{300} = \frac{19}{60} \]

So, \( 0.31\overline{6} = \frac{19}{60} \).

12. Convert \(0.\overline{524}\) to a fraction:

Let \( x = 0.\overline{524} \).
This means \( x = 0.524524524\ldots \).

Multiply both sides by 1000 (since the repeating block has 3 digits):
\[ 1000x = 524.524524524\ldots \]

Now subtract the original equation \( x = 0.524524524\ldots \) from this:
\[ 1000x - x = 524.524524524\ldots - 0.524524524\ldots \]
\[ 999x = 524 \]
\[ x = \frac{524}{999} \]

So, \( 0.\overline{524} = \frac{524}{999} \).

Final Answer:

\[
\boxed{
\begin{array}{ll}
0.\overline{3} = \frac{1}{3} & 0.1\overline{6} = \frac{1}{6} \\
0.\overline{6} = \frac{2}{3} & 0.8\overline{3} = \frac{5}{6} \\
0.\overline{1} = \frac{1}{9} & 0.7\overline{6} = \frac{23}{30} \\
0.\overline{2} = \frac{2}{9} & 0.6\overline{5} = \frac{59}{90} \\
0.\overline{32} = \frac{32}{99} & 0.31\overline{6} = \frac{19}{60} \\
0.\overline{42} = \frac{14}{33} & 0.\overline{524} = \frac{524}{999} \\
\end{array}
}
\]