Solution

(i) $\frac{36}{100}=0.36$

Terminating

(ii) $\frac{1}{11}=0.090909\dots \dots =0.\overline{09}$

Non-terminating repeating

(iii) $4\frac{1}{8}=\frac{33}{8}=4.125$

Terminating

(iv) $\frac{3}{13}=0.230769230769\dots =0.\overline{230769}$

Non-terminating repeating

(v) $\frac{2}{11}=0.18181818$ $=0.\overline{18}$

Non-terminating repeating

(vi) $\frac{329}{400}=0.8225$

Terminating

Solution

Yes. It can be done as follows.

$\begin{array}{rl}& \frac{2}{7}=2\times \frac{1}{7}=2\times 0.\overline{142857}=0.\overline{285714}\\ & \frac{3}{7}=3\times \frac{1}{7}=3\times 0.\overline{142857}=0.\overline{428571}10x=6+x\\ & \frac{4}{7}=4\times \frac{1}{7}=4\times 0.\overline{142857}=0.\overline{571428}\\ & \frac{5}{7}=5\times \frac{1}{7}=5\times 0.\overline{142857}=0.\overline{714285}\\ & \frac{6}{7}=6\times \frac{1}{7}=6\times 0.\overline{142857}=0.\overline{857142}\end{array}$

, where $p$ and $q$ are integers and $q$

$\ne 0$.

Solution

(i)

$0.\bar{6}=0.666\dots$

Let $x=0.666\dots$

$10x=6.666\dots$

$999x=1$

$x=\frac{1}{999}$

Solution

Let $x=0.9999\dots$

$10x=9.9999\dots$

$10x=9+x$

$9x=9x$

=1

Solution

It can be observed that,

$\frac{1}{17}=0.\overline{0588235294117647}$

There are 16 digits in the repeating block of the decimal expansion of $\frac{1}{17}$.

Solution

Terminating decimal expansion will occur when denominator $q$ of rational number $\frac{p}{q}$ is either of $2,4,5,8,10$, and so on…

$\frac{9}{4}=2.25$

$\frac{11}{8}=1.375$

$\frac{27}{5}=5.4$

It can be observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.

Solution

3 numbers whose decimal expansions are non-terminating non-recurring are as follows.

$0.505005000500005000005\dots$

$0.7207200720007200007200000\dots 0.080080008000080000080000008\dots$

Solution

$\frac{5}{7}=0.\overline{714285}$

$\frac{9}{11}=0.\overline{81}$

3 irrational numbers are as follows.

$0.73073007300073000073\dots$

$0.75075007500075000075\dots 0.79079007900079000079\dots$

Solution

As the decimal expansion of this number is non-terminating non-recurring, therefore, it

is an irrational number.

(ii)

$\sqrt{225}=15=\frac{15}{1}$

It is a rational number as it can be represented in $\frac{p}{q}$ form.

(iii) 0.3796

As the decimal expansion of this number is terminating, therefore, it is a rational number.

(iv) $7.478478\dots =7.\overline{478}$

As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.

(v) $1.10100100010000\dots$

As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.