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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.