How to Turn a Repeating Decimal into a Fraction
Create fractions from repeating decimals by following our step-by-step guide.
Published:
Feb 2025
Key takeaways
- • Repeating equals infinite – When a decimal is defined as repeating, it means that the digits following the decimal point recur indefinitely in a specific pattern.
- • Rational expressions – Repeating decimals are rational numbers, and are expressed as a ratio between two integers.
- • Real world applications – Repeating decimals appear frequently in engineering, finance, and science.
Converting repeating decimals into fractions is an important part of understanding how repeating decimals work and how they can be represented in simpler terms. Decimals can’t always express precise numbers clearly. Changing them into fractions allows us to represent these complex numbers more simply.
It may sound difficult, but we promise that changing repeating decimals into fractions is an easy process that you might even find enjoyable! But first, let’s understand what a repeating decimal is and how it connects to the fractions we aim to create.
Definition of Repeating Decimals
Repeating decimals are decimals where digits or groups of digits repeat indefinitely. This could involve a single digit repeating endlessly or a sequence of digits recurring in the same order. When thinking of repeating decimals, it’s important to remember that the numbers following the decimal point follow a specific pattern, whether it be one number repeating itself, or a group of numbers repeating in the same sequence.
This repetition occurs during the long division process when the remainder keeps repeating itself. Essentially, the process continues indefinitely while producing the same remainder, resulting in a perpetual pattern of digits. These decimals are important because they represent precise values of fractions that can’t be expressed with terminating decimals. They enable greater precision in calculations for fields like engineering, science, and finance.
Common Examples of Repeating Decimals
As we just covered, repeating decimals fall into two different categories: single digits following the decimal point that repeat themselves indefinitely, and groups of digits that repeat in a given sequence.
Repeating decimals with only one recurring digit might look something like this—.44444…, a decimal derived from the fraction 49. Decimals with repeating groups of numbers might look like this—.252525…, or 2599. It’s important to note that when dealing with repeating decimals that involve groups of digits after the decimal point, these groups will repeat in specific intervals infinitely.
In the case of the decimal .252525…, the highlighted group will repeat in the specific order, at an interval of two digits.
The reason these decimals are so important to understand lies in how common they are in the real world. Often, the math we use to solve real world problems involves values that don’t divide evenly. Real world calculations are often messy, and the solutions you find, while correct, might not be perfectly neat. Repeating decimals provide us with a way to express these messy calculations in a digestible form.
Basics of Fractions and Their Relation to Decimals
If we’re going to understand the basic functions of repeating decimals, we need to first understand how they fit into the broader picture of fractions and decimals. Let’s take a closer look at fractions.
Definition and Characteristics of Fractions
Fractions represent the relationship between numerators and denominators. In other words, the numerator—or the digit being divided—fits into its denominator—the value used to divide. The numerator is placed above the dividing line in a fraction, while the denominator is below it.
Think of the denominator, the bottom value, as the whole amount, while the numerator on top is just a part of that whole. The connection between this part and its whole forms the foundation of fractions in mathematics.
Types of Fractions
There are three basic types of fractions we’ll be working with in this section: proper, improper, and mixed fractions.
Proper fractions are fractions where the numerator is smaller than the denominator, yielding a value less than 1
Examples:
- 14 = .25
- 34 = .75
- 56 = .83333…
Improper fractions occur when the numerator exceeds the denominator, producing a value greater than 1.
Examples:
- 83 = 2.6666…
- 95 = 1.8
- 103 = 3.3333…
Mixed fractions combine a whole number with a proper fraction. Essentially, mixed fractions represent a whole value with an extra part added on.
Examples:
- 212 = 2.5
- 335 = 3.6
418 = 4.125
Converting Repeating Decimals to Fractions
Now that we understand the types of fractions we’ll be working with, let’s begin learning the steps to convert repeating decimals to fractions.
Identifying the Repeating Part
First, we need to identify the group of digits that forms the repeating sequence. This process is easier for decimals with one repeating digit but requires more attention for repeating groups.
Make sure to identify which groups of digits repeat, and note the interval at which they recur. Once you’ve found the pattern of numbers that continuously repeat, it’s helpful to draw a horizontal line above this sequence of digits. This is known as bar notation, and visually highlights repeating decimal groups.
Setting Up the Equation
To solve for a fraction using a repeating decimal, we start by isolating the repeating decimal itself.
- Identify the repeating digits in your decimal. In the case of .3333…, the recurring digit is 3.
- Next, set the repeating decimal to a variable. For example, let x = .3333… Make sure that only repeating digits come after the decimal point, this is crucial for later!
- Now, multiply both sides of your equation by a power of ten raised to the number of repeating decimals in your problem. Since our equation has one repeating digit, we multiply by 10¹, or simply 10.
- Our equation should now be 10x = 3.333…
- Subtract the equation from step 2 (x = .333…) from the equation in step 3 (10x = 3.333…) to give us 10x – x = 3.333… – .3333…
Solving the Equation
Now that we have our equation set up, we’re ready to solve it!
- 10x – x = 9x, and 3.333… – .333… = 3
- Divide the left side of your equation by the coefficient to isolate your variable.
- 9x9 = 39, we can simplify this to x = 13
Therefore, the repeating decimal of .333… is equal to 13.
Special Cases in Repeating Decimals
There are some special cases of recurring decimals where we’ll need to modify our step-by-step process to correctly calculate our fraction. Here are a few examples of unique problems and solutions.
Non-Repeating Decimals
Non-repeating decimals occur when the digits following the decimal point follow no recognizable pattern, but still run indefinitely. Because of their unending behavior, they’re classified as non-terminating decimals.
Unfortunately, non-repeating decimals can’t be converted into fractions, because they don’t represent rational numbers. Instead, they’re known as irrational numbers, and cannot be expressed as fractions or the ratio of two integers.
Converting Terminating Decimals
Terminating decimals are much easier to convert into fractions than their repeating counterparts. All you have to do is write out the decimal numbers as the numerator of a fraction, and write a power of ten as your denominator.
The power of ten depends on the number of digits in the decimal. Take for example the decimal .1346. There are four decimal numbers, so we’ll express our denominator as 104, or 10,000. Your fraction is then 134610000.
Examples and Step-by-Step Solutions
Let’s convert the following single-digit, multi-digit, and complex repeating decimals into fractions.
Problem 1: Convert .777… into a fraction
- Write your decimal as a function of x
- x = .777…
- Multiply each side by 10 raised to the power of the number of repeating decimals (n)
- 10x = 7.777…
- Subtract the equation from step 1 from the equation found in step 2
- 10x – x – 7.777… – .777…
- Divide each side by the variable’s coefficient
- 9x = 7
- You’re finished!
- x = 79
Problem 2: Convert .15243… into a fraction
- Write your decimal as a function of x
- x = .1524
- Multiply each side by 10n, here we’ll use 104 because we have 4 decimal digits
- 10,000x = 1524.1524…
- Subtract the equation found in step 1 from the equation in step 2
- 10,000x – x = 1524.1524 – .1524…
- Divide each side by the variable’s coefficient
- 9,999x = 1524
- All done!
- x = 15249,999
Problem 3: Convert 1.354… into a fraction
- Write your decimal as a function of x
- x = 1.354…, there are 3 repeating digits (354)
- Multiply each side by 10n, or 103
- 1000x = 1354.354…
- Subtract the equation in step 1 from the equation found in step 2
- 1000x – x = 1354.354 – 1.354
- Simplify and divide
- 999x = 1353
x = 1353999
Conclusion
Repeating decimals occur all over the world of math, so it’s important you understand how to recognize them and their patterns. These terminating and nonterminating decimals play a key role in mathematics, and the step-by-step processes we’ve reviewed here enable you to work with them confidently.
FAQs
A repeating decimal is a decimal in which the digits after the decimal point repeat. This can occur as a single repeating digit or a sequence of repeating digits. These numbers repeat indefinitely following the decimal point.
A decimal is repeating if the digits after the decimal point follow a specific recurring pattern. If only one digit repeats over and over, or if a certain group of digits repeats in a sequence at a given interval, the decimal is considered repeating.
Converting repeating decimals into fractions provides a concise and manageable way to express infinite values. Repeating decimals extend forever, and fractions allow these values to be represented in a condensed form.
Not all decimals are repeating or terminating. There are many decimal values you’ll encounter that are neither. Instead, these values lack any recognizable pattern. These values are known as irrational fractions/decimals.
