Question

How do you convert $ 3.8\bar 3 $ (with a repeating $ 3 $ ) to a fraction?

Answer 1

Hint: In this question, we need to convert $ 3.8\bar 3 $ into fraction. Here, we will consider $ 3.8\bar 3 $ as $ x $ . Then multiply and divide $ 3.8\bar 3 $ by $ 10 $ . Here we will get an equation, mark it as equation (1). Then we will multiply and divide $ 3.8\bar 3 $ by $ 100 $ . By which we will get another equation, mark it as equation (2). And, subtract equation (1) from equation (2), then by evaluating it we will get the required fraction.

Complete step-by-step answer:
In this question, we need to convert $ 3.8\bar 3 $ to a fraction.
Let $ x $ be the fraction that we required.
Here, let us consider $ x = 3.8\bar 3 $ .
Now, multiply and divide $ 3.8\bar 3 $ by $ 10 $ , we have,
 $ x = 3.8333... \times \dfrac{{10}}{{10}} $
Then, $ x = \dfrac{{38.333...}}{{10}} $
Hence, $ 10x = 38.333... $
Let us consider this as equation (1).
Now, let us multiply and divide $ 3.8\bar 3 $ by $ 100 $ , we have,
 $ x = 3.8333... \times \dfrac{{100}}{{100}} $
Then, $ x = \dfrac{{383.333...}}{{100}} $
Hence, $ 100x = 383.333... $
Here, let us consider this as equation (2).
Now, let us subtract equation (1) from equation (2).
Therefore, we have,
 $ 100x - 10x = 383.333... - 38.333... $
Hence, $ 90x = 345 $
So, $ x = \dfrac{{345}}{{90}} $
Therefore, $ x = \dfrac{{23}}{6} $ or $ x = 3\dfrac{5}{6} $ .
Hence, the converted value of $ 3.8\bar 3 $ to a fraction is $ \dfrac{{23}}{6} $ or $ 3\dfrac{5}{6} $ .
So, the correct answer is “$ \dfrac{{23}}{6} $ or $ 3\dfrac{5}{6} $”.

Note: In this question it is important to note that, here we have multiplied and divided $ 3.8\bar 3 $ by $ 10 $ and $ 100 $ respectively, then subtracted both the equations to determine the value of $ x $ as in this question we have a repetition of $ 3 $ in $ 3.8\bar 3 $ . Normally, to convert a decimal to a fraction, place the decimal number over its place value. For example, if we have $ 3.8 $ , the $ 8 $ is in the tenth place, so we place $ 38 $ over $ 10 $ to create the equivalent fraction, i.e., by multiply and dividing by $ 10 $ , we have $ \dfrac{{38}}{{10}} $ . If we have two numbers after the decimal point, then we use $ 100 $ , if there are three then we use $ 1000 $ , etc.