When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. We will explain by means of an example.

If we wish to write 0,3˙0,3˙ in the form abab (where aa and bb are integers) then we would proceed as follows

x
=
0
,
33333
...
10
x
=
3
,
33333
...
multiply
by
10
on
both
sides
9
x
=
3
(
subtracting
the
second
equation
from
the
first
equation
)
x
=
3
9
=
1
3
x
=
0
,
33333
...
10
x
=
3
,
33333
...
multiply
by
10
on
both
sides
9
x
=
3
(
subtracting
the
second
equation
from
the
first
equation
)
x
=
3
9
=
1
3

(6)And another example would be to write
5,
4˙
3˙
2˙
5,
4˙
3˙
2˙
as a rational fraction.

x
=
5
,
432432432
...
1000
x
=
5432
,
432432432
...
multiply by
1000
on both sides
999
x
=
5427
(
subtracting the second equation from the first equation
)
x
=
5427
999
=
201
37
x
=
5
,
432432432
...
1000
x
=
5432
,
432432432
...
multiply by
1000
on both sides
999
x
=
5427
(
subtracting the second equation from the first equation
)
x
=
5427
999
=
201
37

(7)For the first example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the first example there was only one digit (i.e. 3) recurring, while for the second example there were three digits (i.e. 432) recurring.

In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet?

The number of zeros is the same as the number of recurring digits.

Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like
2=1,4142135...2=1,4142135...
cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions instead of decimals.

- Write the following using the repeated decimal notation:
- 0,11111111...0,11111111...
- 0,1212121212...0,1212121212...
- 0,123123123123...0,123123123123...
- 0,11414541454145...0,11414541454145...

Click here for the solution - Write the following in decimal form, using the repeated decimal notation:
- 2323
- 13111311
- 456456
- 219219

Click here for the solution - Write the following decimals in fractional form:
- 0,6333˙0,6333˙
- 5,313131¯5,313131¯
- 0,999999˙0,999999˙

Click here for the solution