Converting Exact Decimals to Fractions

Exact or Non-Repeating Decimals

An exact decimal number is a decimal number that terminates, meaning it has a finite or exact amount of numbers to the right of the decimal point.

Examples

0.25  1.458  98.6  0.123123

All of these are examples of exact decimals. Even though the last example did repeat once, it didn't repeat infinitely many times.

An exact decimal does not have any infinitely repeating patterns, such as

0.\overline{33}  0.\overline{142857}

where the line over the numbers represents an infinitely repeating pattern of those numbers.

Conversion

When we want to convert an exact decimal into a fraction \frac{a}{b}, we must first form a fraction out of that decimal number.

We do this by making the decimal number itself the numerator a and placing b=1 in the denominator.

Example

Let's choose 0.25 as our decimal number. Then

0.25=\frac{0.25}{1}

We get rid of the decimal point in the numerator by moving it to the right the number of places to where it is not needed anymore.

For 0.25, we would move the decimal point to the right two spaces, to the right of the 5

0.25\to 25

Next, we add the same exact number of 0's to the right of the 1 in the denominator as the number of places we moved the decimal point in the numerator.

So we add two 0's to the right of the 1

1\to 100

This process forms an equivalent fraction to our original fraction that does not have the decimal point in the numerator

\frac{0.25}{1}=\frac{25}{100}

After we form the correct fraction to represent the initial decimal, we need to simplify it if possible

\frac{25}{100}=\frac{1}{4}

It helps to remember this rule:

We add the number of 0's to the right of the 1 in the denominator that equals the number of places we moved the decimal point to the right to get rid of it.

Example

Convert 1.504 into a fraction.

First, we divide 1.504 by 1 to form a fraction

1.504=\frac{1.504}{1}

Then move the decimal point right three places and add three 0's to the right of the 1

1.504=\frac{1.504}{1}=\frac{15.04}{10}=\frac{150.4}{100}=\frac{1504}{1000}

Here we showed all of the intermediate steps that show the processes of moving the decimal point to the right and adding the same amount of 0's simultaneously.

Next, simplify

\frac{1504}{1000}=\frac{752}{500}=\frac{376}{250}=\frac{188}{125}

Example

Convert 10.0001 to a fraction

10.0001=\frac{10.0001}{1}=\frac{100001}{10000}

The best Maths tutors available
1st lesson free!
Ayush
5
5 (27 reviews)
Ayush
£90
/h
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£39
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Myriam
5
5 (15 reviews)
Myriam
£20
/h
1st lesson free!
Andrea
5
5 (12 reviews)
Andrea
£40
/h
1st lesson free!
Ayush
5
5 (27 reviews)
Ayush
£90
/h
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£39
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Myriam
5
5 (15 reviews)
Myriam
£20
/h
1st lesson free!
Andrea
5
5 (12 reviews)
Andrea
£40
/h
First Lesson Free>

Converting a Repeating Decimal to Fraction

What if we have a decimal that repeats? How do we find the answer to a problem like that?

Example

Say we have 0.\overline{33}=0.\overline{3}. You may know that this repeating decimal's fractional value is just \frac{1}{3}, but how do we actually get to this answer?

There is a Mathematical trick that we can employ where we just divide the numbers to the right of the decimal by the same amount of 9's placed next to each other, like 9, 99, 999...

0.\overline{3}=\frac{3}{9}=\frac{1}{3}

We have one 3 repeating to the right of the decimal, so we divided it by one 9.

Example

Convert 0.\overline{15} to a decimal

0.\overline{15}=\frac{15}{99}=\frac{5}{33}

We have two numbers 15 in a repeating pattern, so we divided by two 9's to make 99

If there are one or more numbers to the left of the decimal, we still divide by the same amount of 9's as there are to the right of the decimal, but we will need to subtract the number that appears to the left of the decimal from the number that is formed by combining the numbers to the right and to the left of the decimal point.

Example

Convert 1.\overline{3} to a decimal

1.\overline{3}=\frac{13-1}{9}=\frac{12}{9}=\frac{4}3}

The numerator was 13-1=12 because 13 was the number formed by combining all of the numbers and 1 was the number to the left of the decimal point, so we had to subtract it from 13, which results in 12. Then we divided by one 9 because one number 3 was repeating.

Here is the rule for a decimal that repeats after an initial amount of non-repeating decimals

1.1\overline{3}=\frac{113-11}{90}=\frac{102}{90}=\frac{17}{15}

We subtract 11 from 113 because that is the total number of digits that don't repeat.

We divide by one 9 because of the repeating pattern of 3's and add a 0 to the right for the one non-repeating number to the right of the decimal point 1.

Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 5.00/5 - 1 vote(s)
Loading...

Patrick