Question

Write the recurring decimal $0.6\overline{3}$ as a fraction in its lowest terms.

Answer 1

Hint: We need to find the value of $0.6\overline{3}$ as a fraction. Firstly, we assume the value of $x$ equal to $0.6\overline{3}$ and find the value of $10x$ and $100x$ .Then, we subtract $100x$ , $10x$ and simplify the expression to get the desired result.

Complete step-by-step solution:
We are given a repeating decimal and the repeating digit in the decimal is 3. We need to express the decimal as a fraction in the simplest form.
Let us consider the value of the variable $x=0.6333...$
We will be solving the given question by subtracting the terms $100x$ and $10x$ to express the decimal as a fraction.
Fractions, in mathematics, are used to represent the portion or the part of the entire or whole thing. They are generally represented as follows,
$\Rightarrow {\displaystyle \frac{a}{b}}$
Here,
$a$ is the numerator of the fraction
$b$ is the denominator of the fraction
For Example:
$\Rightarrow {\displaystyle \frac{1}{2}},{\displaystyle \frac{2}{5}}$
Decimals, in mathematics, are numbers whose whole number part and fractional part are separated by a decimal point.
According to our question,
We need to express the repeating decimal as a fraction.
According to the question,
$\Rightarrow x=0.6333...$
From the above, we can see that only a single digit in the decimal is repeating.
We need to find the value of $10x$ by multiplying the value of $x$ and 10.
Applying the same, we get,
$\Rightarrow 10\times x=10\times (0.6333...)$
Simplifying the above equation, we get,
$\therefore 10x=6.333...$
We need to find the value of $100x$ by multiplying the value of $x$ and 100.
Applying the same, we get,
$\Rightarrow 100\times x=100\times (0.6333...)$
Simplifying the above equation, we get,
$\therefore 100x=63.33...$
Now, we need to subtract $100x$ and $10x$ .
Subtracting the terms, we get,
$\Rightarrow 100x-10x=(63.33...)-(6.333...)$
Simplifying the above equation, we get,
$\Rightarrow 90x=57.00...$
We need to isolate the variable $x$ to find its value.
Shifting the number 90 to the other side of the equation, we get,
$\Rightarrow x=\frac{(57.00...)}{90}$
Cancelling out the common factors, we get,
$\therefore x={\displaystyle \frac{19}{30}}$

Note: The given question can be solved alternatively as follows,
The formula to convert any repeating decimal to a fraction is given by
$\Rightarrow {\displaystyle \frac{(\text{Decimal}\times F)-\left(\text{Non-repeating part of decimal number}\right)}{D}}$
The value of $F$ is 10 if one digit is repeating in decimal, 100 if two digits are repeating in a decimal.
The value of $D$ is 9 if one digit is repeating in decimal, 99 if two digits are repeating in a decimal.
Here,
Decimal: $0.6\overline{3}$
F: 10
Non-repeating part of decimal number: 0.6
D: 9
Substituting the same, we get,
$\Rightarrow {\displaystyle \frac{(0.63\times 10)-\left(0.6\right)}{9}}$
Simplifying the above expression,
$\Rightarrow {\displaystyle \frac{6.3-0.6}{9}}$
$\Rightarrow {\displaystyle \frac{5.7}{9}}$
Cancelling out the common factors,
$\Rightarrow {\displaystyle \frac{19}{30}}$