Rapid Math
Converting Repeating Decimals to Fractions
Any rational umber can be expressed as a fraction. Repeating Decimals and
Terminal Decimals are all Rational Numbers, so they can be converted into
1
fractions. We can use repeating decimal patterns directly, such as = 0. 1̅ ,
1
2
9
9
̅ = , 0. 2̅ =
𝑠𝑜 0. 1
̅=
, 0. 3
3
9
9
1
= 3 , … 𝑒𝑡𝑐. here I will show you how to convert some
other type of repeating decimals to fractions by using repeating decimal patterns
indirectly.
̅ to fraction,
Example 1, converting 𝟎. 𝟖𝟐
Step 1, separate terminal decimal portion and repeating decimal portion:
0.82̅ = 0.8 + 0.02̅,
Step 2, convert each portion to its fraction form,
8
4
1
2
2
1
0.8 = 10 = 5 , 0.02̅ = 0.1 × 0. 2̅ = 10 × 9 = 90 = 45
Step 3, add those fractions together and done,
4
5
+
1
45
=
36
45
+
1
45
=
37
45
37
So, 0.82̅ =
45
̅̅̅to fraction,
Example 2, converting 𝟑. 𝟏𝟓
3.25̅ = 3 + 0.2 + 0.05̅ = 3 +
2
1 5
2
5
18 5
23
+
× =3 +
=3 +
=3
10 10 9
10 90
90 90
90
̅̅̅̅ to fraction,
Example 3, converting 𝟏. 𝟖𝟑𝟕
̅̅̅̅ = 1.2 + 0.037
̅̅̅̅ = 1
1.237
Academic Success Centre
2
1 37
2
37
198 37
235
47
+
×
=1 +
=1
+
=1
=1
10 10 99
10 990
990 990
990
198
Compiled by Ming Xu
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