Question

The sum of \[0.\overline{3}\] and \[0.\overline{4}\] is: -
(a) \[\dfrac{7}{10}\]
(b) \[\dfrac{7}{9}\]
(c) \[\dfrac{7}{99}\]
(d) \[\dfrac{7}{11}\]

Answer 1

Hint: First, convert the given repeating decimal numbers into fractions. To do this, select \[0.\overline{3}\] and write it as 0.3333…. Multiply it with 10 by assuming x = 0.3333…… Now, subtract x from 10x and divide both the sides with 9 to get the value of x in fractional form. Similarly, do the same process to find the value of \[0.\overline{4}\] in fractional form by assuming it as y = 0.444….. Finally, take the sum of x and y obtained in the fractional form to get the answer.

Complete step by step answer:
Here, we have been provided with the decimal numbers \[0.\overline{3}\] and \[0.\overline{4}\] and we are asked to find the sum of these numbers.
Now, we can see that the given numbers are repeating and non -terminating. So, we cannot add them up to infinite places after the decimal point. Therefore, we need to convert them in the fractional form in the first step. So, we have,
Let us assume, \[0.\overline{3}\] = x.
\[\Rightarrow x=0.\overline{3}\]
Removing the bar sign we have,
\[\Rightarrow \] x = 0.3333….. – (1)
Multiplying both sides with 10, we get,
\[\Rightarrow \] 10x = 3.3333…… - (2)
Subtracting equation (1) from equation (2), we have,
\[\Rightarrow \] 9x = 3.000….
\[\Rightarrow \] 9x = 3
\[\Rightarrow x=\dfrac{3}{9}\] - (3)
Now, let us assume \[0.\overline{4}\] = y
\[\Rightarrow \] y = \[0.\overline{4}\]
Removing the bar sign, we get,
\[\Rightarrow \] y = 0.444…. – (4)
Multiplying both sides with 10, we get,
\[\Rightarrow \] 10y = 4.444…. – (5)
Subtracting equation (4) from equation (5), we have,
\[\Rightarrow \] 9y = 4.000….
\[\Rightarrow y=\dfrac{4}{9}\] - (6)
Now, equation (3) and (6) represents the numbers x and y in fractional form respectively. We have to find the sum of \[0.\overline{3}\] and \[0.\overline{4}\], i.e. x and y. So, we have,
\[\Rightarrow 0.\overline{3}+0.\overline{4}=x+y\]
Substituting the values of x and y from equation (3) and (6), we get,
\[\Rightarrow 0.\overline{3}+0.\overline{4}=\dfrac{3}{9}+\dfrac{4}{9}\]
Taking L.C.M in the R.H.S, we have,
\[\begin{align}
  & \Rightarrow 0.\overline{3}+0.\overline{4}=\dfrac{4+3}{9} \\
 & \Rightarrow 0.\overline{3}+0.\overline{4}=\dfrac{7}{9} \\
\end{align}\]
Hence, option (b) is the required answer.

Note:
One may note that these given numbers were rational numbers and that is why we were able to convert them in fractional form. You may see that here the digits were repeating continuously, that is why we multiplied them by 10. If the numbers were like \[0.\overline{43}\] which will repeat after an interval of one digit then to convert it in the fractional form we would have multiplied it with 100. Remember that we cannot add them up to infinite places and so this conversion is needed.