Question

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

Answer 1

Hint: We need to find the value of $0.6\bar{3}$ as a fraction. Firstly, we assume the value of $x$ equal to $0.6\bar{3}$ 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,
$\Rightarrow \dfrac{a}{b}$
Here,
$a$ is the numerator of the fraction
$b$ is the denominator of the fraction
For Example:
$\Rightarrow \dfrac{1}{2},\dfrac{2}{5}$
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,
$\Rightarrow 10\times x=10\times \left( 0.6333... \right)$
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,
$\Rightarrow 100\times x=100\times \left( 0.6333... \right)$
Simplifying the above equation, we get,
$\therefore 100x=63.33...$
Now, we need to subtract $100x\;$ and $10x\;$ .
Subtracting the terms, we get,
$\Rightarrow 100x-10x=\left( 63.33... \right)-\left( 6.333... \right)$
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,
$\Rightarrow x=\frac{\left( 57.00... \right)}{90}$
Cancelling out the common factors, we get,
$\therefore x=\dfrac{19}{30}$

Note: The given question can be solved alternatively as follows,
The formula to convert any repeating decimal to a fraction is given by
$\Rightarrow \dfrac{\left( \text{Decimal}\times F \right)-\left( \text{Non-repeating part of decimal number} \right)}{D}$
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: $0.6\bar{3}$
F: 10
Non-repeating part of decimal number: 0.6
D: 9
Substituting the same, we get,
$\Rightarrow \dfrac{\left( 0.63\times 10 \right)-\left( 0.6 \right)}{9}$
Simplifying the above expression,
$\Rightarrow \dfrac{6.3-0.6}{9}$
$\Rightarrow \dfrac{5.7}{9}$
Cancelling out the common factors,
$\Rightarrow \dfrac{19}{30}$