Question

How do you convert \[0.254\](4 repeating) as a fraction?

Answer 1

Hint: A fraction consists of a numerator and a denominator. Converting a decimal number into a fractional one would require removing the repeating part by subtracting appropriately. We can start solving by considering 10x equal to the given number, i.e. 0.254 and then proceed to find the value of x.

Complete step by step answer:
The given question can be represented as \[0.25\bar{4}\], where the bar over the number 4 denotes that 4 is repeating.
For eg- if a number is given as \[0.12\bar{3}\], this denotes that the number is \[0.12333...\](3 is repeating)
According to the question, we have to convert the given decimal into a fraction
Firstly, let us assign it to a variable, that is
Let \[x=0.254.....................(1)\]
Now the next step is important, since we have only one number as repeating we will multiply both the sides by 10.
So if in a question we have two repeating numbers then we will multiply both sides by 100. Therefore, in short we can say, the number of digits repeating, n will require \[{{10}^{n}}\]to be multiplied on both sides.
Carrying on with our calculations, we will multiply both sides of equation 1 by 10, we get
\[10x=2.544.....................(2)\]
Now we need to subtract equation 2 by equation 1 that is
\[eq(2)-eq(1)\]
\[10x-x=2.544-0.254\]
\[\Rightarrow 9x=2.29\]
\[\Rightarrow x=\dfrac{2.29}{9}\]
We got the value of x in fraction but it still has a decimal point so we will multiply and divide x by 100. Since we are multiplying and dividing by the same number so it does not change the value of x.
\[\Rightarrow x=\dfrac{2.29}{9}\times \dfrac{100}{100}\]
\[x=\dfrac{229}{900}\]

Note: The number of repeating digits have an important role for equation 2, this is valid for any question of this type. If a mistake is made in this part, the answer can get wrong. And also the final answer should not have any decimal adjust it by multiplying and dividing an appropriate number.