Question

How do you convert $\dfrac{3}{11}$ to a decimal equivalent using long division?

Answer 1

Hint:We first try to explain the proper fraction. We use variables to express the condition between those representations. Then we apply a long division to express the proper fraction part into decimal. The decimal is non-terminating recurring decimal. We express it in its proper mathematical form.

Complete step by step solution:
The given fraction $\dfrac{3}{11}$ is a proper fraction. proper fractions are those fractions who have lesser value in numerator than the denominator. We need to convert it into decimal.
Now we solve our fraction $\dfrac{3}{11}$.
We express it in regular long division processes. The denominator is the divisor. The numerator is the dividend. The quotient will be the integer of the mixed fraction. The remainder will be the numerator of the proper fraction.
$11\overset{0.272}{\overline{\left){\begin{align}
& 30 \\
& \underline{22} \\
& 80 \\
& \underline{77} \\
& 30 \\
& \underline{22} \\
& 8 \\
\end{align}}\right.}}$
The decimal is non-terminating recurring decimal. The value remainder is always 3 and 8 and these numbers keep coming in that order as long as we keep dividing. So, we express this mathematical form in such a way that we only have to use the repeating digit only once.
So, $\dfrac{3}{11}=0.27272727....=0.\overline{27}$. The overline tells us which digits get repeated.
Therefore, the decimal form of the proper fraction $\dfrac{3}{11}$ is $0.\overline{27}$.

Note: The given decimal number is a representation of non-terminating recurring decimals. These types of decimal numbers are rational numbers.
To understand the process better we take another example of $2.45\overline{74}$.
The fractional form of the decimal form will be $2.45\overline{74}=\dfrac{24574-
245}{9900}=\dfrac{24329}{9900}$.
There are two recurring and two non-recurring digits in that number after decimal. That’s why we used two 9s and two 0s in the denominator.