# Repeating Decimal Calculator + Online Solver With Free Steps

The **Repeating Decimal Calculator** is used to solve repeating decimal numbers into their fraction forms. This is helpful as **Repeating Decimal Numbers** are infinitely long and they are difficult to express in their decimal form, so expressing them in a **Fraction Form **can provide detailed information about their true value.

## What Is a Repeating Decimal Calculator?

**The Repeating Decimal Calculator is an online calculator which can convert repeating decimal numbers into their corresponding fractions.**

This **Calculator** is very helpful as converting fractions to decimals is easy but converting decimals to fractions can be challenging.

And this **Calculator** does it all in your browser and needs nothing but a problem to solve.

## How to Use the Repeating Decimal Calculator?

To use the **Repeating Decimal Calculator**, you must place the decimal value in the input box and press the button, and you shall have your results. It is a very intuitive and easy-to-use calculator.

The step-by-step guide is as follows:

### Step 1

Enter your repeating decimal number in the input box.

### Step 2

Press the button labeled, “Submit”.

### Step 3

And you have your solution presented to you in a new window. In case you want to solve more problems of the same nature, you can enter them in the new window.

## How Does the Repeating Decimal Calculator Work?

The **Repeating Decimal Calculator** works by taking in a repeating decimal number and then solving it to find the corresponding fraction for it. We are aware that fractions and decimal numbers are easily **Interchangeable**, but most one is used to convert a fraction into a decimal.

Thus, converting a decimal number to a fraction can be challenging but there is always a way. Now, before we move towards the method of **Converting** said repeating decimal numbers to fractions, let’s go into detail about **Repeating Decimal Numbers** themselves.

### Repeating Decimal Numbers

**Repeating Decimal Numbers** are therefore **non-terminating** decimal numbers, which means that the values after the decimal will carry on till **Infinity**. And the major difference from common **non-terminating** decimal numbers here is the recurring nature of its decimal values, where one or more numbers will present themselves in a **Repeating Fashion**.

These can’t be **Zeros**.

### Convert Repeating Decimal Numbers to Fractions

Now, the method of solving such a problem involving almost a **Reversed Process** of decimal to fraction conversion uses **Algebra** of all things. So, the** Technique** used is that we take our repeating decimal number as the variable x, and we multiply certain values to it.

Now, let there be a **Repeating Decimal Number** x, and let n be the number of repeating digits in the decimal values of this number. We shall **Multiply** this number by $10^n$ first and get:

\[ 10^n x = y \]

Hence, this will result in a **Mathematical Value** y, then we take that value and** Subtract** from it the number $10^{n-1}$ multiplied with the original x giving us a value z. This is done so that we can **Eliminate** the decimal part of the resulting value and hence get an integer:

**$10^n$ x – $10^{n-1}$ x = y – z = a**

Here, a is the resulting value from y – z, and this value is intended to have no decimal values attached to it, so it has to be an **Integer**. And now we can solve this algebraic expression as follows:

**($10^n$ – $10^{n-1}$) x = a**

\[ x = \frac{a}{10^n – 10^{n-1}}\]

And thus, we can have the final result which would be a **Fraction** representing the value x we started from. Therefore, it’s the equivalent fraction to our **Repeating Decimal Number** we had hoped to find.

## Solved Examples

Now, let’s get a better understanding of the method at hand by going and looking at some solved examples.

### Example 1

Consider the repeating decimal number 0.555555, and find the fraction equivalent of it.

### Solution

We begin by first setting up a **Notation** for this number, this is done here:

** x = 0.555555 **

Now, we move forward by counting the number of **Repeating Values** in the decimal of this number. This number comes out to be 1 as there is only 5 which is repeating till **Infinity**. So, now we use the value we learned about above $ 10^n $, and multiply our x with it:

**n = 1, $ 10^n$ = $10^1 $ = 10 **

**10 x = 5.555555 **

Here, we have our **Algebraic Equation** set up, now we must solve for the $10 ^{n-1}$ value, and that can be seen done as follows:

**n -1 = 1 – 1 = 0, $10^{n-1}$ = $10^0$ = 1 **

We subtract 1x on both sides:

**10x – x = 5.555555 – 0.555555 = 5**

Therefore,

**9x = 5, x = $\frac{5}{9}$**

Hence, we have our fraction solution.

### Example 2

Consider the given repeating decimal number as 1.042424242, and calculate the fraction equivalent for it.

### Solution

We first start by using the appropriate **Notation** for this problem:

**x = 1.042424242 **

Moving forward, we count the quantity of the **Repeating Values** present in our x. We can see that the repeating numbers here are 2 which are 42 repeating till **infinity**. Now, we will use the $10^n$ for this number, but one **Important Thing** to notice is that the first three numbers after the decimal are 042 which are unique so, we will take an n = 3 for this case:

**n = 3, $10^n$ = $10^3$ = 1000 **

**1000 x = 1042.42424242 **

Then we follow that up with the $10^{n-1}$ but given the nature of this problem, to **Eliminate** the decimal values we have to use $10^{n-2}$:

** n -2 = 3 – 2 = 1, $10^{n-1}$ = $10^1$ = 10 **

Subtracting 10x on both sides looks like:

**1000x – 10x = 1042.42424242 – 10.42424242 = 1032 **

Hence,

**990x = 1032, x = $\frac{1032}{990}$**

Finally, we have our solution.