Question

How do you convert \[\dfrac{2}{{11}}\] to a decimal?

Answer 1

Hint:We will rewrite \[\dfrac{2}{{11}}\] in a form where the numerator is greater than the denominator. To do this, we will multiply and divide by \[10\]. Then, we will perform the division operation and divide the resulting decimal value by \[10\].

Complete step by step solution:
We have to find the decimal expansion of \[\dfrac{2}{{11}}\].
Here, we observe that the denominator is greater than the numerator.
We will multiply both the numerator and the denominator of \[\dfrac{2}{{11}}\] by \[10\]. This gives us
\[\dfrac{{2 \times 10}}{{11 \times 10}} = \dfrac{{20}}{{110}}\]
Now, we have \[110 = 11 \times 10\].
So, instead of writing the denominator as a product of 11 and 10, we can divide \[\dfrac{{20}}{{11}}\] by \[10\] to get the same value as \[\dfrac{{20}}{{110}}\].
This means we are rewriting \[\dfrac{{20}}{{110}}\] as
\[ \Rightarrow \dfrac{{20}}{{110}} = \dfrac{{\left( {\dfrac{{20}}{{11}}} \right)}}{{10}}\]
Here, the numerator is \[\dfrac{{20}}{{11}}\]. We can find the decimal expansion of \[\dfrac{{20}}{{11}}\] easily by normal division.
Now, the multiple of 11 that is closer to 20 is 11 itself.
So, we will multiply 11 by 1 to get 11.
Subtracting 11 from 20, we get the remainder as 9, which is lesser than 11.
So, we insert a decimal point in the quotient and affix a zero next to 9 in the remainder.
Now, the multiple of 11 that is closer to 90 is \[11 \times 8 = 88\].
So, we multiply 11 by 8 to get 88.
Subtracting 88 from 90, we get 2, which is again lesser than 11. So we affix a zero and the same procedure continues.
Hence, we have
\[\dfrac{{20}}{{11}} = 1.8181...\]
To get the value of \[\dfrac{{20}}{{110}}\], we have to divide the above value by 10.
Therefore, we shift the decimal point one place to the left to get \[0.1818...\]

Hence, we have obtained the value of \[\dfrac{2}{{11}}\] as \[\dfrac{2}{{11}} = 0.1818...\].

Note: To solve the given problem, we may adopt an alternate method:
Rewrite \[\dfrac{2}{{11}}\] as \[\dfrac{2}{{11}} = 2 \times \dfrac{1}{{11}}\].
Then, we will adopt the above method to find the value of \[\dfrac{1}{{11}}\]. Therefore, we get
\[\dfrac{1}{{11}} = 0.0909...\]
Now, multiplying this result by 2, we get
\[\dfrac{2}{{11}} = 2 \times 0.0909 = 0.1818...\].