Question

How do you convert 0.12345 (12345 repeating) to a fraction?

Answer 1

Hint: This question is from the topic of algebra. In this question, we will place a bar above the number 12345 after the decimal. After that we will multiply the number 10 having the power of 5, so that it will be easy to solve this question. And after that we will do the further process to solve this question.

Complete step by step answer:
Let us solve this question.
The question is asking us to convert the term 0.12345 (12345 repeating) into the fractional value.
The term 0.12345 where 12345 is repeating can also be written as
\[0.\overline{12345}\]
Let the term \[0.\overline{12345}\] be x.
The term \[0.\overline{12345}\] can also be written as \[0.12345\overline{12345}\].
As we can see that 5 digits are repeating after decimal. And the digits are 12345.
 So, we can write
\[x=0.12345\overline{12345}\]
As we can see that 5 digits are repeating after the decimal. So, we will multiply 10 to the power of 5 to the both sides of the equation.
Hence, after multiplying \[{{10}^{5}}\] to the both side of equation, we get
\[{{10}^{5}}\times x={{10}^{5}}\times 0.12345\overline{12345}\]
The above equation can also be written as
\[\Rightarrow 100000\times x=100000\times 0.12345\overline{12345}\]the above equation can also be written as
\[\Rightarrow 100000x=12345.\overline{12345}\]
The above equation can also be written as
\[\Rightarrow 100000x=12345+0.\overline{12345}\]
As we know that the value of x is \[0.\overline{12345}\].
So, we can write the above equation as
\[\Rightarrow 100000x=12345+x\]
By taking the term x to the left side of equation, we get
\[\Rightarrow 100000x-x=12345\]
The above equation can also be written as
\[\Rightarrow 99999x=12345\]
The above equation can also be written as
\[\Rightarrow x=\dfrac{12345}{99999}\]
Now, we will check the prime factorization of 12345 and 99999
The prime factorization of 12345 will be \[3\times 5\times 823\]
And the prime factorization of \[99999\] will be \[3\times 3\times 41\times 271\].
Hence, we can write
\[\Rightarrow x=\dfrac{3\times 5\times 823}{3\times 3\times 41\times 271}\]
After cancelling out 3, the above equation can also be written as
\[\Rightarrow x=\dfrac{5\times 823}{3\times 41\times 271}\]
\[\Rightarrow x=\dfrac{4115}{33333}\]
Hence the fractional value of \[0.\overline{12345}\] is \[\dfrac{4115}{33333}\].

Note:
We should have a better knowledge in the chapter algebra to solve this type of question. We can solve this question by a different method.
We will multiply the number (which we have to convert in fractional value) by \[{{10}^{n}}-1\] where n is the number of digits after the decimal.
So, the number is \[0.\overline{12345}\] and the number of digits after the decimal is 5.
Hence, we will multiply \[{{10}^{5}}-1\] that is (100000-1) with \[0.\overline{12345}\] to get an integer.
So, we can write
\[0.\overline{12345}\left( {{10}^{5}}-1 \right)=0.\overline{12345}\left( 100000-1 \right)=12345.\overline{12345}-0.\overline{12345}=12345\]
We can simply write the above equation as
\[0.\overline{12345}\left( {{10}^{5}}-1 \right)=12345\]
Dividing \[{{10}^{5}}-1\] both sides we get
\[\Rightarrow 0.\overline{12345}=\dfrac{12345}{{{10}^{5}}-1}=\dfrac{12345}{99999}\]
As we have found above that \[\dfrac{12345}{99999}=\dfrac{4115}{33333}\]
So, we can write
\[\Rightarrow 0.\overline{12345}=\dfrac{4115}{33333}\]