Question

Express each of the following as a fraction in simplest form: \[2.\overline{4}\] .

Answer 1

Hint: These types of problems are not quite straight forward but can be solved quickly once we understand all the core concepts behind the problem. Solving this particular problem requires some previous knowledge of chapters like fractions, number systems, recurring and non-recurring decimals. Here we are given a bar over the last digit of the decimal, which represents that this decimal number is non-terminating. To solve such problems, we first of all need to initialize the decimal to a certain variable and then proceed with it.

Complete step-by-step solution:
Now we start off with the solution to the given problem by initializing the decimal number to some variable as,
\[x=2.\overline{4}\]
This can also be rewritten as, \[x=2.444444.......\] . Now we simply multiply with \[10\] on both the sides of the equation to get,
\[\Rightarrow 10x=24.44444.......\]
Now what we do is, subtract \[x\] from both the sides of the equation of the above intermediate equation to find,
\[\begin{align}
  & \Rightarrow 10x-x=24.44444.......-2.44444..... \\
 & \Rightarrow 9x=22 \\
\end{align}\]
Now from the above intermediate we can very easily find the value of \[x\] , which is nothing but the initial value that we had assumed and the required answer to the problem. We now divide both the sides of the equation by \[9\] to get,
\[\Rightarrow x=\dfrac{22}{9}\]
This is the required answer to the problem.

Note: This type of problems require some thorough knowledge of chapters like number systems, recurring and non-recurring decimals. We need to analyse these type of problems very carefully as they are not very easy to predict. We also need to be very careful with the calculations of the problem or else it may lead to an error and finally result in a wrong answer.