Question

Write the following in decimal form and say what kind of decimal expansion each has?
(i) \[\dfrac{{36}}{{100}}\]
(ii) \[\dfrac{1}{{11}}\]
(iii) \[4\dfrac{1}{8}\]
(iv) \[\dfrac{3}{{13}}\]
(v) \[\dfrac{2}{{11}}\]
(vi) \[\dfrac{{329}}{{400}}\]

Answer 1

Hint: In above question, we are given six rational numbers. They are of different types. We have to check what kind of decimal expansions each have. In order to approach the solution, first we have to convert these fractional numbers into their decimal forms and then see what kind of decimal expansions they have by the digits after the decimal.

Complete step by step solution:
Given rational numbers are \[\dfrac{{36}}{{100}}\] , \[\dfrac{1}{{11}}\] , \[4\dfrac{1}{8}\] , \[\dfrac{3}{{13}}\] , \[\dfrac{2}{{11}}\] , \[\dfrac{{329}}{{400}}\] .
We have to find their decimal expansions.
There are three types of decimal expansions they are as follows:
1) Terminating: When the digits after the decimal point terminate or vanish after a finite number of digits then it is called terminating decimal expansion. Ex- \[3.25\] .
2) Non-terminating but repeating: When the digits after the decimal point do not terminate, i.e. they continue to infinite numbers of digits but in a repeating manner of finite digits then they are called non-terminating but repeating decimal expansions. Ex- \[6.6666...\] . We can use a bar to denote the repeating digits such as \[6.\overline {66} \] .
3) Non-terminating non-repeating: When the digits after the decimal point do not terminate and also do not continue in a repetitive manner then they are called non-terminating non repeating decimal expansions. Ex- \[4.2324252627...\] .
Now we have,
(i) \[\dfrac{{36}}{{100}}\]
We can write \[\dfrac{{36}}{{100}}\] as
\[ \Rightarrow \dfrac{{36}}{{100}} = 0.36\]
Hence it is terminating decimal expansion.

(ii) \[\dfrac{1}{{11}}\]
We can \[\dfrac{1}{{11}}\] as
\[ \Rightarrow \dfrac{1}{{11}} = 0.090909090...\]
Or,
\[ \Rightarrow \dfrac{1}{{11}} = 0.\overline {09} \]
Hence, it is non-terminating but repeating.

(iii) \[4\dfrac{1}{8}\]
We can write \[4\dfrac{1}{8}\] as
\[ \Rightarrow 4\dfrac{1}{8} = \dfrac{{33}}{8}\]
Or,
\[ \Rightarrow 4\dfrac{1}{8} = 4.125\]
Hence, it is terminating.

(iv) \[\dfrac{3}{{13}}\]
We can write \[\dfrac{3}{{13}}\] as,
\[ \Rightarrow \dfrac{3}{{13}} = 0.23076923...\]
Hence, it is non-terminating non-repeating decimal expansion.

(v) \[\dfrac{2}{{11}}\]
We can write \[\dfrac{2}{{11}}\] as,
\[ \Rightarrow \dfrac{2}{{11}} = 0.181818...\]
Or,
\[ \Rightarrow \dfrac{2}{{11}} = 0.\overline {18} \]
Hence, it is non-terminating but repeating.

(vi) \[\dfrac{{329}}{{400}}\]
We can write \[\dfrac{{329}}{{400}}\] as,
\[ \Rightarrow \dfrac{{329}}{{400}} = 0.8225\]
Hence, it is terminating decimal expansion.

Note: The decimal expansions of type 1 and 2, i.e. terminating and non-terminating but repeating, can be conversely converted into fractional forms but not the third kind i.e. non-terminating non repeating.
Examples:
1) \[0.2\] can be written as \[\dfrac{2}{{10}}\] .
2) \[0.\overline {33} \] can be written as \[\dfrac{1}{9}\] .
3) \[0.1211221112222...\] cannot be written as a rational number and as a result it is an irrational number.