Question

How do you convert \[\dfrac{7}{9}\] to a decimal using long division?

Answer 1

Hint: To solve this we should know about the basics of division.
Division: The division is the method of distributing a group of things into a number of equal parts. Here we have to use one more concept of division that is long division which means keep dividing after decimal the number unless the division is terminated or starts repeating the same pattern.

Complete step by step solution:
We have to convert \[\dfrac{7}{9}\] to a decimal number.
So, we divide $7$ by $9$ .
As we see $7$ is smaller than $9$ . So, the quotient will be less than $1$ .
We normally start dividing \[\dfrac{7}{9}\] .
 $9\left| \!{\overline {\,
 7 \,}} \right. $
But $7$ is smaller than $9$. So, we have to take zero after $7$ and write decimal at after zero in quotient. As shown below,
 $
  \,\,0. \\
  9\left| \!{\overline {\,
 {70} \,}} \right. \\
 $
We will proceed further after each step add zero after remainder as we know we can write zero after decimal without changing in number. Repeat the same step and keep dividing.
 \[
  \,\,0.7 \\
  9\left| \!{\overline {\,
 {70} \,}} \right. \\
   - \,63 \\
  \,\,\,\,\,07 \\
  \,\,\,\,\,\,\,\, \\
 \]
As, we get $7$ as a reminder then we will put zero behind it as after decimal we can put zero without changing its magnitude and divide it by $9$ .
 \[
  \,\,0.77 \\
  9\left| \!{\overline {\,
 {70} \,}} \right. \\
   - \,63 \\
  \,\,\,\,\,\,70 \\
  \,\, - \,63 \\
  \,\,\,\,\,\,0\,7 \\
 \]
We again get $7$ as a reminder. We put zero after $7$ and further divide it by $9$ .
 \[
  \,\,0.777 \\
  9\left| \!{\overline {\,
 {70} \,}} \right. \\
   - \,63 \\
  \,\,\,\,\,\,70 \\
  \,\, - \,63 \\
  \,\,\,\,\,\,\,\,\,70 \\
  \,\,\,\,\, - 63 \\
  \,\,\,\,\,\,\,\,\,07 \\
 \]
We again get $7$ as a reminder.
So, as we keep on dividing it will go for infinite times.
As we found the division is infinitely long but we have found $7$ is repeating.
Hence, we can write it as, \[\dfrac{7}{9} = 0.777777\]
We can rewrite it as \[\dfrac{7}{9} = 0.\bar 7\] as $7$ is repeating so we place the bar over repeating terms.
So, the correct answer is “\[ 0.\bar 7\] ”.

Note: Division is a basic mathematical operation. There are four basic mathematical operations: addition, subtraction, multiplication and division. Division is traditionally considered as left-associative. That is, if there are multiple divisions in a row, then the order of calculation goes from left to right. Division is written in a format of dividend over the divisor with a horizontal line which is called a fraction bar.