Question

Express $\left( {\dfrac{2}{{11}}} \right)$ in decimal form.

Answer 1

Hint: In the given question, we are required to convert a given fraction into decimal. The fraction is having numerator and denominator. So, we first simplify the fraction and bring it in simplest form. Then, in order to convert it in decimal we will perform the division. The problem requires basic knowledge of long division methods and representation in decimal form.

Complete step-by-step solution:
The given question requires us to convert the given fraction into its simplest form by cancelling out the common factors in numerator and denominator and then expressing it as decimal representation.
So, to convert the given fraction into its simplest form, we factorise the numerator as well as denominator and look for the common factors.
We have, $2 = 2 \times 1$ and $11 = 11 \times 1$.
Therefore, there is no common factor between numerator and denominator. Hence, the fraction can be written as $\left( {\dfrac{2}{{11}}} \right)$ in its simplest form where numerator and denominator have no common factor between them.

Now, to convert the fraction into the decimal representation, we divide the numerator by the denominator until we get zero as the remainder.
On dividing $2$ by $11$, we get a repeating and recurring decimal expansion.
Now, when we are dividing $2$ by $11$, we have to place a decimal point as $2$ is less than $11$ and hence cannot be divided without placing a decimal point. Now, as we have placed a decimal point in the quotient, we can draw a zero and hence now we have to divide $20$ by $11$. So, we get, $1$ as quotient and $9$ as remainder.
Now, we draw another zero and hence we have to now divide $90$ by $11$. So, we get $8$ as a quotient and $2$ as remainder. Therefore, this process goes on and gives us a repeating and recurring decimal expansion.

So, we have, $\left( {\dfrac{2}{{11}}} \right) = 0.181818....$

Note: The division of the numerator by the denominator can be a cumbersome task as it involves thorough knowledge of algebraic rules and long division method. Care should be taken while doing the same and proceeding with the process to convert a fraction into decimal. The decimal expansion of a rational number in $\dfrac{p}{q}$ form can be terminating or recurring in nature.