Question

How do you write $3.5$ repeating as a fraction in simplest form?

Answer 1

Hint: $3.5$ repeating means $3.5555....$ .To write a repeating decimal as a fraction in simplest form, we need to suppose that decimal as $x$ and then multiply it by $10$ and then subtract $x$ from the result. On solving the obtained equation, we will get the fraction form of the given repeating decimal.

Complete step-by-step solution:
We are given a repeating decimal and we need to express the decimal as a fraction in simplest form.
Now, we know that a repeating decimal is a rational number and it can be converted to fraction form.
First of all, let us suppose this repeating decimal is $x$ .
 $\Rightarrow x=3.5555....------\left(1\right)$
Now, to eliminate the digits after the decimal point, we need to multiply $x$ with $10$ .
Multiplying equation $\left(1\right)$ with $10$ , we get
 $\Rightarrow 10x=35.5555....-----\left(2\right)$
Now, we have to subtract $x$ from the result.
So, equation $\left(2\right)-$ $\left(1\right)$ gives
 $$\begin{gathered} \Rightarrow 10x-x=35.5555...-3.5555... \\ \Rightarrow 9x=32 \\ \Rightarrow x={\displaystyle \frac{32}{9}} \end{gathered}$$
This is our final answer.

Hence, we have written $3.5$ repeating as ${\displaystyle \frac{32}{9}}$ a fraction in simplest form.

Note: We can solve this question using an alternate method also. In this method, we can use a direct formula. The formula is
 $\Rightarrow {\displaystyle \frac{\left(Decimal\times F\right)-\left(Nonrepeatingpart\right)}{D}}$
Here,
 $F=10$ , if only one digit is repeating.
 $F=100$ , if two digits are repeating.
 $D=9$ , if one digit is repeating.
 $D=99$ , if two digits are repeating.
In our question, only one digit that is $5$ is repeating.
So, $F=10,D=9$ and the non-repeating part is 3.
Putting this values in the formula we get,
Fraction form of $3.555...$ $={\displaystyle \frac{\left(3.5\times 10\right)-3}{9}}$
$$\begin{gathered} ={\displaystyle \frac{35-3}{9}} \\ ={\displaystyle \frac{32}{9}} \end{gathered}$$