Question

Convert 3.9(9 repeating) to a fraction?

Answer 1

Hint: We write the original number with repeating digit after the decimal place. Assume the given number as a variable. Multiply the number and the variable with 10. Subtract one equation from another and calculate the value of the variable.
Repeating a decimal number is a number where the digit after the decimal occurs again and again an infinite number of times. Example: 2.77777…, 9.111111… etc.

Complete step by step solution:
We are given a number 3.9 with 9 repeating. This means the number is 3.99999…
Let us assume the number equal to a variable, say x
Let \[x = 3.9999...\] … (1)
Now we multiply equation (1) by 10 i.e. we multiply 10 to both sides of the equation
\[ \Rightarrow 10 \times x = 10 \times 3.9999...\]
Shift the decimal point to one place on the right hand side in the number 3.999…
\[ \Rightarrow 10x = 39.999...\] … (2)
Now we subtract equation (1) from equation (2)
\[ \Rightarrow 9x = 36\]
Divide both sides of the equation by 9
\[ \Rightarrow \dfrac{{9x}}{9} = \dfrac{{36}}{9}\]
Cancel same factors from numerator and denominator on both sides of the equation
\[ \Rightarrow x = 4\]
We can write right hand side of the equation in fraction form as
\[ \Rightarrow x = \dfrac{4}{1}\]
Since we know from equation (1) \[x = 3.9999...\]
So, we can write the number 3.999… in fraction form as \[\dfrac{4}{1}\].
\[\therefore \]The fraction form of 3.9(9 repeating) is \[\dfrac{4}{1}\].

Note: Many students make the mistake of writing fraction forms of 3.999… by just writing the number 399999… in the numerator and 10 with power n in the denominator. Keep in mind we can convert decimal to fraction if we have a fixed number of digits after the decimal, here we don’t know how many digits occur after decimal.