Math worksheet for converting repeating decimals to fractions.

Here are the solutions for converting the recurring decimals into fractions.

To solve these, we use a standard method:
1. Let $x$ equal the recurring decimal.
2. Multiply $x$ by a power of 10 (like 10, 100, 1000) to move the decimal point so that the repeating part lines up.
3. Subtract the original equation from the new one to remove the repeating decimals.
4. Solve for $x$ and simplify the fraction.

1) $0.\overline{3}$
Let $x = 0.333...$
Multiply by 10: $10x = 3.333...$
Subtract $x$: $9x = 3$
$x = \frac{3}{9} = \frac{1}{3}$

2) $1.\overline{4}$
Let $x = 1.444...$
Multiply by 10: $10x = 14.444...$
Subtract $x$: $9x = 13$
$x = \frac{13}{9}$ (or $1 \frac{4}{9}$)

3) $0.\overline{57}$
Let $x = 0.5757...$
Multiply by 100 (since 2 digits repeat): $100x = 57.5757...$
Subtract $x$: $99x = 57$
$x = \frac{57}{99}$
Divide top and bottom by 3: $\frac{19}{33}$

4) $0.\overline{001}$
Let $x = 0.001001...$
Multiply by 1000 (since 3 digits repeat): $1000x = 1.001001...$
Subtract $x$: $999x = 1$
$x = \frac{1}{999}$

5) $0.\overline{53}$
Let $x = 0.5353...$
Multiply by 100: $100x = 53.5353...$
Subtract $x$: $99x = 53$
$x = \frac{53}{99}$

6) $2.\overline{93}$
Let $x = 2.9393...$
Multiply by 100: $100x = 293.9393...$
Subtract $x$: $99x = 291$
$x = \frac{291}{99}$
Divide top and bottom by 3: $\frac{97}{33}$ (or $2 \frac{31}{33}$)

7) $18.\overline{48}$
Let $x = 18.4848...$
Multiply by 100: $100x = 1848.4848...$
Subtract $x$: $99x = 1830$
$x = \frac{1830}{99}$
Divide top and bottom by 3: $\frac{610}{33}$ (or $18 \frac{16}{33}$)

8) $0.2\overline{35}$
This one is tricky because the '2' doesn't repeat, only '35' does.
Let $x = 0.23535...$
Multiply by 10 to move the non-repeating part: $10x = 2.3535...$
Multiply by 1000 to move one full cycle of the repeat: $1000x = 235.3535...$
Subtract the first from the second: $990x = 233$
$x = \frac{233}{990}$

9) $0.00\overline{32}$
Let $x = 0.003232...$
Multiply by 100 to pass the non-repeating zeros: $100x = 0.3232...$
Multiply by 10000 to pass the repeating part: $10000x = 32.3232...$
Subtract: $9900x = 32$
$x = \frac{32}{9900}$
Divide by 4: $\frac{8}{2475}$

10) $0.\overline{9}$
Let $x = 0.999...$
Multiply by 10: $10x = 9.999...$
Subtract $x$: $9x = 9$
$x = \frac{9}{9} = 1$

11) $2.4\overline{178}$
Let $x = 2.4178178...$
Multiply by 10: $10x = 24.178178...$
Multiply by 10000: $10000x = 24178.178178...$
Subtract: $9990x = 24154$
$x = \frac{24154}{9990}$
Divide by 2: $\frac{12077}{4995}$ (or $2 \frac{2087}{4995}$)

12) $2.\overline{36}$
Let $x = 2.3636...$
Multiply by 100: $100x = 236.3636...$
Subtract $x$: $99x = 234$
$x = \frac{234}{99}$
Divide by 9: $\frac{26}{11}$ (or $2 \frac{4}{11}$)

13) $1.\overline{323}$
Let $x = 1.323323...$
Multiply by 1000: $1000x = 1323.323...$
Subtract $x$: $999x = 1322$
$x = \frac{1322}{999}$ (or $1 \frac{323}{999}$)

14) $0.3\overline{178}$
Let $x = 0.3178178...$
Multiply by 10: $10x = 3.178178...$
Multiply by 10000: $10000x = 3178.178178...$
Subtract: $9990x = 3175$
Divide by 5: $x = \frac{635}{1998}$

15) $0.4\overline{07}$
Let $x = 0.40707...$
Multiply by 10: $10x = 4.0707...$
Multiply by 1000: $1000x = 407.0707...$
Subtract: $990x = 403$
$x = \frac{403}{990}$

16) $32.12\overline{35}$
Let $x = 32.123535...$
Multiply by 100: $100x = 3212.3535...$
Multiply by 10000: $10000x = 321235.3535...$
Subtract: $9900x = 318023$
$x = \frac{318023}{9900}$ (or $32 \frac{1223}{9900}$)

Final Answer:
1) $\frac{1}{3}$
2) $\frac{13}{9}$
3) $\frac{19}{33}$
4) $\frac{1}{999}$
5) $\frac{53}{99}$
6) $\frac{97}{33}$
7) $\frac{610}{33}$
8) $\frac{233}{990}$
9) $\frac{8}{2475}$
10) $1$
11) $\frac{12077}{4995}$
12) $\frac{26}{11}$
13) $\frac{1322}{999}$
14) $\frac{635}{1998}$
15) $\frac{403}{990}$
16) $\frac{318023}{9900}$