18.5 More Complex Repeating
Decimals to Fractions
8. NS.1 Know that there are numbers that are not rational, and
approximate them by rational numbers. Know that numbers that are not
rational are called irrational. Understand informally that every number has
a decimal expansion; for rational numbers show that the decimal
expansion repeats eventually, and convert a decimal expansion which
repeats eventually into a rational number.
WARM-UP
Convert to a fraction.
1) .4
2) .81
Evaluate the expressions.
3) 10 ÷ .2
10
4)
.2
5) 6 ÷ .03
6)
6
.03
More Complex Repeating Decimals
to Fractions
Do all repeating decimals end up over 9,
99, and 999?
x = .16
10x = 1.66
− x = .16
9x = 1.5
9
9
15 ÷15 1
x=
=
90 ÷15 6
NOTES
Remember that we always need to simplify our fractions.
Examples
Convert to a fraction.
.12
.34
NOTES
For mixed numbers its easiest to leave out the whole
number until the end.
Examples
Convert to a fraction.
7.2
4.81
NOTES
The number of repeating digits determines whether we
multiply by 10, 100, or 1000 not their location -- but
repeating digits still need to line up.
Concept Check
Set-up the equation to convert to a fraction.
.03
.02
NOTES
We can clear decimals from fractions the same as
equations by moving the decimal point in both
locations.
Examples
Convert to a fraction.
.03
10x = .33
.02
10x = .22
− x = .03
− x = .02
9x = .3
9x = .2
EXAMPLES
Convert to a fraction.
.07
Convert to a fraction.
(hint there is a one in the
numerator)
.083
PRACTICE
Convert the decimal to a fraction.
1.8
.56
PRACTICE
Convert the decimal to a fraction.
.06
FINAL QUESTION
Convert to a fraction (hint there is a one in the
numerator).
.142857