Question

How do you convert $0.53434.34.......$ as a fraction?

Answer 1

Hint: We are given a decimal expression as $0.53434.34.......$it has a repeat term we are asked to convert it into a fraction. We will first put bar (-) on the term which is being repeated then we will follow steps that respond to convert the decimal to fraction. We can also cross check our solution by again simplifying the fraction into the decimal form.

Complete step-by-step solution:
We are given a decimal $0.53434.34.......$here $34$ is keep on repeating we know when bar (-) is placed over the term it symbolises that term is repeating so we can write our decimal as $0.53434.34.......=0.5\overline{34}$
Now we have to change $0.5\overline{34}$ into fraction we will follow step by step interia to change the above decimals to the fraction.
Step 1: We name the decimal as x
So, we have
$x=0.5\overline{34}\,\,\,.........(i)$
Step: 2 We will multiply eq (i) by the number \[\text{10}\,\,\text{or}\,\text{100}\,\,\text{or}\,\,\text{1000}\]depending upon the number that lie between decimal and the bar.
In $x=0.5\overline{34}\,\,$, there is $1$ term between decimal and bar, so we multiply equation (i) by $10$
So, we get,
$10\times x=0.5\overline{34}\times 10$
Simplifying we get
$10x=5.\overline{34}......(ii)$
Step: 3 now we multiply by \[\text{10}\,\,\text{or}\,\text{100}\,\,\text{or}\,\,\text{1000}\] depending upon the number of term and are under bar.
As in $10x=5.\overline{34}$there are $2$ term under the bar so we will multiply equation (ii) by $100\,\text{(2}\,\text{Zero}\,\text{as}\,\text{number}\,\text{under}\,\text{bar)}$
So,
Equation (ii) e get,
$10x\times 100=5.\overline{34}\times 100$
We get,
$1000x=534.\overline{34}......(iii)$
Step: 4 Subtract (ii) from (iii)
So we subtract $10x=5.\overline{34}$ from
\[\begin{align}
  & 1000x=534.\overline{34} \\
 & 10x=5.\overline{34} \\
 & \overline{990x=529.00} \\
\end{align}\]
So, we get
\[990x=529.00..........(iv)\]
Ste: 5 Divide the equation (iv) by coefficient of x
As coefficient of equation (iv) is $990$
So, we divide $990x=529$by $990$
So, we get,
$x=\dfrac{529}{990}$
as initial we know $x=0.5\overline{34}\,\,$
So, we get
$x=0.5\overline{34}\,\,=\dfrac{529}{990}$
So, fraction form of $0.5\overline{34}\,\,$is $\dfrac{529}{990}$

Note: Remember if we put decimal as $x=0.5\overline{34}\,\,$ and we multiply by $1000$(as it has $3$decimal places ) and get $1000x=534.\overline{34}$, if we subtract them two will not get the correct solution as both right first term has repeated term from 2nd place while on the 2nd equation $1000x=534.\overline{34}$ repeated addition and subtraction won’t work on there and the solution will be incorrect. We need to follow the given step.