Rational Numbers

EXERCISE-9.1

P1. List five rational numbers between:

(i) − 1 and 0

(ii) − 2 and − 1

(iii) $\frac{\mathbf{–}\mathbf{4}}{\mathbf{5}}\mathbf{ }\mathbf{and}\mathbf{ }\frac{\mathbf{–}\mathbf{2}}{\mathbf{3}}$

(iv)  $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{ }\mathbf{and}\mathbf{ }\frac{\mathbf{2}}{\mathbf{3}}$

Sol. (i)−1 and 0

$\frac{-1}{10},\frac{-1}{20},\frac{-1}{30},\frac{-1}{40},\frac{-1}{50}$

(ii) −2 and −1

$-2=\frac{-12}{6} \mathrm{and}-1=\frac{-6}{6}$

Five rational numbers are $\frac{-11}{6},\frac{-10}{6},\frac{-9}{6},\frac{-8}{6},\frac{-7}{6}$

(iii)  $\frac{-4}{5} \mathrm{and} \frac{-2}{3}$

$\frac{-4}{5}=\frac{-4\times 9}{5\times 9}=\frac{-36}{45}$ and $\frac{-2}{3}=\frac{-2\times 15}{3\times 15}=\frac{-30}{45}$

Five rational numbers are $\frac{-35}{45},\frac{-34}{45},\frac{-33}{45},\frac{-32}{45},\frac{-31}{45}$

(iv) $\frac{1}{2} \mathrm{and} \frac{2}{3}$

$\frac{1}{2}=\frac{1\times 18}{2\times 18}=\frac{18}{36}$ and $\frac{2}{3}=\frac{2\times 12}{3\times 12}=\frac{24}{36}$

Five rational numbers are $\frac{19}{36},\frac{20}{36},\frac{21}{36},\frac{22}{36},\frac{23}{36}$

P2. Write four more rational numbers in each of the following patterns:

(i)  $\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20},\dots$

(ii)  $\frac{-1}{4},\frac{-2}{8},\frac{-3}{12}\dots ..$

(iii)  $\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24}\dots$

(iv)  $\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9}\dots$

Sol. (i)  $\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20}\dots$

$\frac{-3}{5},\frac{-3\times 2}{5\times 2},\frac{-3\times 3}{5\times 3},\frac{-3\times 4}{5\times 4}....$

It can be observed that the numerator is a multiple of 3 while the denominator is a multiple of 5 and as we increase them further, these multiples are increasing. Therefore, the next four rational numbers in this pattern are

$\frac{-3\times 5}{5\times 5},\frac{-3\times 6}{5\times 6},\frac{-3\times 7}{5\times 7},\frac{-3\times 8}{5\times 8}\dots$

$\frac{-15}{25},\frac{-18}{30},\frac{-21}{35},\frac{-24}{40}\dots$

(ii) $\frac{-1}{4},\frac{-2}{8},\frac{-3}{12}\dots$

$\frac{-1}{4},\frac{-1\times 2}{4\times 2},\frac{-1\times 3}{4\times 3}\dots$

The next four rational numbers in this pattern are

$\frac{-1\times 4}{4\times 4},\frac{-1\times 5}{4\times 5},\frac{-1\times 6}{4\times 6},\frac{-1\times 7}{4\times 7}\dots$

$\frac{-4}{16},\frac{-5}{20},\frac{-6}{24},\frac{-7}{28}\dots$

(iii) $\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24}\dots$

$\frac{-1}{6},\frac{1\times 2}{-12\times 2},\frac{1\times 3}{-6\times 3},\frac{1\times 4}{-6\times 4}\dots$

The next four rational numbers in this pattern are

$\frac{1\times 5}{-6\times 5},\frac{1\times 6}{-6\times 6},\frac{1\times 7}{-6\times 7},\frac{1\times 8}{-6\times 8}\dots$

$\frac{5}{-30},\frac{6}{-36},\frac{7}{-42},\frac{8}{-48}\dots$

(iv) $\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9}\dots$

$\frac{2}{3},\frac{2}{-3},\frac{2\times 2}{-3\times 2},\frac{2\times 3}{-3\times 3}\dots$

The next four rational numbers in this pattern are

$\begin{array}{l}\frac{2\times 4}{-3\times 4},\frac{2\times 5}{-3\times 5},\frac{2\times 6}{-3\times 3},\frac{2\times 7}{-3\times 7}\\ \frac{8}{-12},\frac{10}{-15},\frac{12}{-18},\frac{14}{-21}\dots \end{array}$

P3. Give four rational numbers equivalent to:

$\left(i\right)\mathbf{ }\frac{\mathbf{–}\mathbf{2}}{\mathbf{7}}$

$\left(\mathrm{ii}\right)\mathbf{ }\frac{\mathbf{5}}{\mathbf{–}\mathbf{3}}$

$\left(\mathrm{iii}\right)\mathbf{ }\frac{\mathbf{4}}{\mathbf{9}}$

Sol. (i) $\frac{–2}{7}$

Four rational numbers are $\frac{-2\times 2}{7\times 2},\frac{-2\times 3}{7\times 3},\frac{-2\times 4}{7\times 4},\frac{-2\times 5}{7\times 5}$

$\frac{-4}{14},\frac{-6}{21},\frac{-8}{28},\frac{-10}{35}$

(ii) $\frac{5}{–3}$

Four rational numbers are $\frac{5\times 2}{-3\times 2},\frac{5\times 3}{-3\times 3},\frac{5\times 4}{-3\times 4},\frac{5\times 5}{-3\times 5}$

$\frac{10}{-6},\frac{15}{-9},\frac{20}{-12},\frac{25}{-15}$

(iii) $\frac{4}{9}$

Four rational numbers are $\frac{4\times 2}{9\times 2},\frac{4\times 3}{9\times 3},\frac{4\times 4}{9\times 4},\frac{4\times 5}{9\times 5}$

$\frac{8}{18},\frac{12}{27},\frac{16}{36},\frac{20}{45}$

P4. Draw the number line and represent the following rational numbers on it:

$\left(i\right)\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}$

$\left(\mathrm{ii}\right)\mathbf{ }\frac{\mathbf{–}\mathbf{5}}{\mathbf{8}}$

$\left(\mathrm{iii}\right)\mathbf{ }\frac{\mathbf{–}\mathbf{7}}{\mathbf{4}}$

$\left(\mathrm{iv}\right)\mathbf{ }\frac{\mathbf{7}}{\mathbf{8}}$

Sol. (i) $\frac{3}{4}$

This fraction represents 3 parts out of 4 equal parts. Therefore, each space between two integers on number line must be divided into 4 equal parts . $\frac{3}{4}$can be represented as

(ii) $\frac{–5}{8}$

This fraction represents 5 parts out of 8 equal parts. Negative sign represents that it is on the negative side of number line. Therefore, each space between two integers on number line must be divided into 8 equal parts. $\frac{–5}{8}$ can be represented as

$\left(\mathrm{iii}\right) \frac{-7}{4}=-1\frac{3}{4}$

This fraction represents 1 full part and 3 parts out of 4 equal parts. Negative sign represents that it is on the negative side of number line. Therefore, each space between two integers on number line must be divided into 4 equal parts. $\frac{–7}{4}$ can be represented as

$\left(\mathrm{iv}\right) \frac{7}{8}$

This fraction represents 7 parts out of 8 equal parts. Therefore, each space between two integers on number line must be divided into 8 equal parts. $\frac{7}{8}$ can be represented as

P5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

Sol. Distance between U and T = 1 unit

It is divided into 3 equal parts.

$\mathrm{TR}=\mathrm{RS}=\mathrm{SU}=\frac{1}{3}$

$\mathrm{R}=-1-\frac{1}{3}=-\frac{3}{3}-\frac{1}{3}=\frac{-4}{3}$

$\mathrm{S}=-1-\frac{2}{3}=-\frac{3}{3}-\frac{2}{3}=-\frac{5}{3}$

Similarly,

AB = 1 unit

It is divided into 3 equal parts.

$P=2+\frac{1}{3}=\frac{6}{3}+\frac{1}{3}=\frac{7}{3}$

$Q=2+\frac{2}{3}=\frac{6}{3}+\frac{2}{3}=\frac{8}{3}$

P6. Which of the following pairs represent the same rational number?

$\left(\mathrm{i}\right) \frac{-7}{21} \mathrm{and} \frac{3}{9}$

$\left(\mathrm{ii}\right) \frac{-16}{20} \mathrm{and} \frac{20}{-25}$

$\left(\mathrm{iii}\right) \frac{-2}{-3} \mathrm{and} \frac{2}{3}$

$\left(\mathrm{iv}\right) \frac{-3}{5} \mathrm{and} \frac{-12}{20}$

$\left(\mathrm{v}\right) \frac{8}{-5} \mathrm{and} \frac{-24}{15}$

$\left(\mathrm{vi}\right) \frac{1}{3} \mathrm{and} \frac{-1}{9}$

$\left(\mathrm{vii}\right) \frac{-5}{-9} \mathrm{and} \frac{5}{-9}$

Sol. (i) $\frac{-7}{21} \mathrm{and} \frac{3}{9}$

$\frac{-7}{21}=\frac{-1}{3}$

$\frac{3}{9}=\frac{1}{3}$

As, $\frac{-1}{3}\ne \frac{1}{3}$ therefore, it does not represent same rational numbers.

(ii) $\frac{-16}{20} \mathrm{and} \frac{20}{-25}$

$\frac{-16}{20}=\frac{-4}{5}$

$\frac{-20}{25}=\frac{-4}{5}$

Therefore, it represents same rational numbers.

(iii) $\frac{-2}{-3}\mathrm{and}\frac{2}{3}$

$\frac{-2}{-3}=\frac{2}{3}$

Therefore, it represents same rational numbers.

(iv) $\frac{-3}{5}=\frac{-12}{20}$

$\frac{-12}{20}=\frac{-3}{5}$

Therefore, it represents same rational numbers.

(v) $\frac{8}{-5} \mathrm{and} \frac{-24}{15}$

$\frac{-24}{15}=\frac{-8}{5}$

$\frac{8}{-5}=\frac{8}{5}$

Therefore, it represents same rational numbers.

(vi) $\frac{1}{3} \mathrm{and} \frac{-1}{9}$

As $\frac{1}{3}\ne \frac{-1}{9},$ therefore, it does not represent same rational numbers.

(vii) $\frac{-5}{-9} \mathrm{and} \frac{5}{-9}$

$\frac{-5}{-9}=\frac{5}{9}$

As $\frac{5}{9}\ne \frac{-5}{9}$ , therefore, it does not represent same rational numbers.

P7. Rewrite the following rational numbers in the simplest form:

$\left(i\right)\frac{-8}{6}$

$\left(ii\right) \frac{25}{45}$

$\left(iii\right) \frac{–44}{72}$

$\left(iv\right) \frac{–8}{10}$

Sol.  $\left(\mathrm{i}\right) \frac{-8}{6}=\frac{-4\times 2}{3\times 2}=\frac{-4}{3}$

$\left(\mathrm{ii}\right) \frac{25}{45}=\frac{5\times 5}{9\times 5}=\frac{5}{9}$

$\left(\mathrm{iii}\right) \frac{-44}{72}=\frac{-11\times 4}{18\times 4}=\frac{-11}{18}$

$\left(\mathrm{iv}\right) \frac{-8}{10}=\frac{-4\times 2}{5\times 2}=\frac{-4}{5}$

P8. Fill in the boxes with the correct symbol out of >, <, and =

$\left(\mathrm{i}\right) \frac{-5}{7}\square \frac{2}{3}$

$\left(\mathrm{ii}\right) \frac{-4}{5}\square \frac{-5}{7}$

$\left(\mathrm{iii}\right) \frac{-7}{8}\square \frac{14}{-16}$

$\left(\mathrm{iv}\right) \frac{-8}{5}\square \frac{-7}{4}$

$\left(\mathrm{v}\right) \frac{1}{-3}\square \frac{-1}{4}$

$\left(\mathrm{vi}\right) \frac{5}{-11}\square \frac{-5}{11}$

$\left(\mathrm{vii}\right) 0 \square \frac{-7}{6}$

Sol. $\left(\mathrm{i}\right) \frac{-5}{7}=\frac{-5\times 3}{7\times 3}=\frac{-15}{21}$

$\frac{2}{3}=\frac{2\times 7}{3\times 7}=\frac{14}{21}$

As $-15<14$

Therefore,$\frac{-5}{7}\boxed{<}\frac{2}{3}$

(ii) $\frac{-4}{5}=\frac{-4\times 7}{5\times 7}=\frac{-28}{35}$

$\frac{-5}{7}=\frac{-5\times 5}{7\times 5}=\frac{-25}{35}$

As $-28<-25$

Therefore , $\frac{-4}{5}\boxed{<}\frac{-5}{7}$

(iii) Here, $\frac{14}{16}=\frac{7\times 2}{-8\times 2}=\frac{7}{-8}=\frac{-7}{8}$

Therefore, $\frac{-7}{8}\boxed{=}\frac{14}{-16}$

(iv) $\frac{-8}{5}=\frac{-8\times 4}{5\times 4}=\frac{-32}{20}$

$\frac{-7}{4}=\frac{-7\times 5}{4\times 5}=\frac{-35}{20}$

As $-32>-35 ,$

Therefore, $\frac{-8}{5}\boxed{>}\frac{-7}{4}$

(v) $\frac{-1}{3}=\frac{-1\times 4}{3\times 4}=\frac{-4}{12}$

$\frac{-1}{4}=\frac{-1\times 3}{4\times 3}=\frac{-3}{12}$

As $-4<-3,$

Therefore, $\frac{1}{-3}\boxed{<}\frac{-1}{4}$

$\left(\mathrm{vi}\right) \frac{5}{-11}\boxed{=}\frac{-5}{11}$

$\left(\mathrm{vii}\right) 0 \boxed{>} \frac{-7}{6}$

P9. Which is greater in each of the following?

$\left(\mathrm{i}\right) \frac{2}{3},\frac{5}{2}$

$\left(\mathrm{ii}\right) \frac{-5}{6},\frac{-4}{3}$

$\left(\mathrm{iii}\right) \frac{-3}{4},\frac{2}{-3}$

$\left(\mathrm{iv}\right) \frac{-1}{4},\frac{1}{4}$

$\left(\mathrm{v}\right) -3\frac{2}{7},-3\frac{4}{5}$

Sol. (i) $\frac{2}{3},\frac{5}{2}$

By converting these into like fractions,

$\frac{2}{3}=\frac{2\times 2}{3\times 2}=\frac{4}{6}$

$\frac{5}{2}=\frac{5\times 3}{2\times 3}=\frac{15}{6}$

As 15 > 4, therefore, $\frac{5}{2}$ is greater.

$\left(\mathrm{ii}\right) \frac{-5}{6},\frac{-4}{3}$

$\frac{-4}{3}=\frac{-4\times 2}{3\times 2}=\frac{-8}{6}$

As -5 > -8, therefore, $\frac{–5}{6}$ is greater.

$\left(\mathrm{iii}\right) \frac{-3}{4},\frac{2}{-3} \mathrm{or} \frac{-3}{4},\frac{-2}{3}$

By converting these into like fractions,

$\frac{-3}{4}=\frac{-3\times 3}{4\times 3}=\frac{-9}{12}$

$\frac{-2}{3}=\frac{-2\times 4}{3\times 4}=\frac{-8}{12}$

As – 8 > – 9, therefore, $\frac{–2}{3}$ is greater.

$\left(\mathrm{iv}\right) \frac{-1}{4},\frac{1}{4}$

$\frac{1}{4}>\frac{-1}{4}$

$\left(\mathrm{v}\right) -3\frac{2}{7},-3\frac{4}{5}$

$\frac{-23}{7},\frac{-19}{5}$

By converting these into like fractions,

$\frac{-23}{7}=\frac{-23\times 5}{7\times 5}=\frac{-115}{35}$

$\frac{-19}{5}=\frac{-19\times 7}{5\times 7}=\frac{-133}{35}$

As – 115 > – 133, therefore, $-3\frac{2}{7}$ is greater.

P10. Write the following rational numbers in ascending order:

$\left(\mathrm{i}\right) \frac{-3}{5},\frac{-2}{5},\frac{-1}{5}$

$\left(\mathrm{ii}\right) \frac{-1}{3},\frac{-2}{9},\frac{-4}{3}$

$\left(\mathrm{iii}\right) \frac{-3}{7},\frac{-3}{2},\frac{-3}{4}$

Sol. $\left(\mathrm{i}\right) \frac{-3}{5},\frac{-2}{5},\frac{-1}{5}$

As −3 < −2 < −1,

$\therefore \frac{-3}{5}<\frac{-2}{5}<\frac{-1}{5}$

$\left(\mathrm{ii}\right) \frac{-1}{3},\frac{-2}{9},\frac{-4}{3}$

By converting these into like fractions,

$\frac{-1\times 3}{3\times 3},\frac{-2}{9},\frac{-4\times 3}{3\times 3}$

$\frac{-3}{9},\frac{-2}{9},\frac{-12}{9}$

As −12 < −3 < −2,

$\therefore \frac{-4}{3}<\frac{-1}{3}<\frac{-2}{9}$

$\left(\mathrm{iii}\right) \frac{-3}{7},\frac{-3}{2},\frac{-3}{4}$

By converting these into like fractions,

$\frac{-3\times 4}{7\times 4},\frac{-3\times 14}{2\times 14},\frac{-3\times 7}{4\times 7}$

$\frac{-12}{28},\frac{-42}{28},\frac{-21}{28}$

As −42 < −21 < −12,

$\therefore \frac{-3}{2}<\frac{-3}{4}<\frac{-3}{7}$

EXERCISE-9.2

P1. Find the sum:

(i) $\frac{4}{5}+\left(\frac{-11}{4}\right)$

(ii) $\frac{5}{3}+\frac{3}{5}$

(iii) $\frac{-9}{10}+\frac{22}{15}$

(iv) $\frac{-3}{11}+\frac{5}{9}$

(v) $\frac{-8}{19}+\frac{(-2)}{57}$

(vi) $\frac{-2}{3}+0$

(vii) $-2\frac{1}{3}+4\frac{3}{5}$

Sol. (i) $\frac{4}{5}+\left(\frac{-11}{4}\right)$

L.C.M of 5 and 4 is 20

$\frac{4}{5}+\left(\frac{-11}{4}\right)$= $\frac{4\times 4}{5\times 4}–\frac{11\times 5}{4\times 5}= \frac{16}{20}–\frac{55}{20}$

$\frac{16–55}{20}=\frac{–39}{20}$

(ii) $\frac{5}{3}+\frac{3}{5}$

L.C.M of 3 and 5 is 15.

$\frac{5}{3}+\frac{3}{5}=\frac{5\times 5}{3\times 5}+\frac{3\times 3}{5\times 3}=\frac{25}{15}+\frac{9}{15}$

$\frac{25+9}{15}=\frac{34}{15}$

(iii) $\frac{-9}{10}+\frac{22}{15}$

L.C.M of 10 and 15 is 30.

$\frac{-9}{10}+\frac{22}{15}=\frac{-9\times 3}{10\times 3}+\frac{22\times 2}{15\times 2}$

$\frac{-27}{30}+\frac{44}{30}=\frac{-27+44}{30}=\frac{17}{30}$

(iv) $\frac{-3}{-11}+\frac{5}{9}=\frac{3}{11}+\frac{5}{9}$

L.C.M of 11 and 9 is 99.

$\frac{3}{11}+\frac{5}{9}=\frac{3\times 9}{11\times 9}+\frac{5\times 11}{9\times 11}$

$\frac{27}{99}+\frac{55}{99}=\frac{27+55}{99}=\frac{82}{99}$

(v) $\frac{-8}{19}+\frac{(-2)}{57}=-\frac{8}{19}-\frac{2}{57}$

L.C.M of 19 and 57 is 57.

$\frac{-8}{19}-\frac{2}{57}=-\frac{8\times 3}{19\times 3}-\frac{2}{57}$

$-\frac{24}{57}-\frac{2}{57}=\frac{-24-2}{57}=\frac{-26}{57}$

$\left(\mathrm{vi}\right) \frac{-2}{3}+0=\frac{-2}{3}$

$\left(\mathrm{vii}\right) -2\frac{1}{2}+4\frac{3}{5}=\frac{-7}{3}+\frac{23}{5}$

L.C.M of 3 and 5 is 15.

$\frac{-7}{3}+\frac{23}{5}=\frac{-7\times 5}{3\times 5}+\frac{23\times 3}{5\times 3}$

$\frac{-35}{15}+\frac{69}{15}=\frac{-35+69}{15}=\frac{34}{15}$

P2. Find

$\left(\mathrm{i}\right) \frac{7}{24}-\frac{17}{36}$

$\left(\mathrm{ii}\right) \frac{5}{63}-\left(\frac{-6}{21}\right)$

$\left(\mathrm{iii}\right) \frac{-6}{13}-\left(\frac{-7}{15}\right)$

$\left(\mathrm{iv}\right) \frac{-3}{8}-\frac{7}{11}$

$\left(\mathrm{v}\right) -2\frac{1}{9}-6$

Sol. (i) $\frac{7}{24}-\frac{17}{36}$

L.C.M of 24 and 36 is 72.

$\frac{7}{24}-\frac{17}{36}=\frac{7\times 3}{24\times 3}-\frac{17\times 2}{36\times 2}$

$=\frac{21}{72}-\frac{34}{72}=\frac{21-34}{72}=\frac{-13}{72}$

(ii) $\frac{5}{63}-\left(\frac{-6}{21}\right)=\frac{5}{63}+\frac{2}{7}$

L.C.M of 63 and 7 is 63.

$\frac{5}{63}+\frac{2}{7}=\frac{5}{63}+\frac{2\times 9}{7\times 9}=\frac{5}{63}+\frac{18}{63}$

$\frac{5+18}{63}=\frac{23}{63}$

(iii) $\frac{-6}{13}-\left(\frac{-7}{15}\right)=\frac{-6}{13}+\frac{7}{15}$

L.C.M of 13 and 15 is 195.

$\frac{-6}{13}+\frac{7}{15}=\frac{-6\times 15}{13\times 15}+\frac{7\times 13}{15\times 13}$

$=\frac{-90}{195}+\frac{91}{195}=\frac{-90+91}{195}$

(iv) $\frac{-3}{8}-\frac{7}{11}$

L.C.M of 8 and 11 is 88.

$\frac{-3}{8}-\frac{7}{11}=-\frac{3\times 11}{8\times 11}-\frac{7\times 8}{11\times 8}$

= $-\frac{33}{88}-\frac{56}{88}=\frac{-33-56}{88}=\frac{-89}{88}$

(v) $-2\frac{1}{9}-6=-\frac{19}{9}-\frac{6}{1}$

L.C.M of 9 and 1 is 9.

$-\frac{19}{9}-\frac{6}{1}=-\frac{19}{9}-\frac{6\times 9}{1\times 9}=-\frac{19}{9}-\frac{54}{9}$

= $\frac{-19-57}{9}=\frac{-73}{9}$

P3. Find the product:

(i) $\frac{9}{2}\times \left(\frac{-7}{4}\right)$

(ii) $\frac{3}{10}\times (-9)$

(iii) $\frac{-6}{5}\times \frac{9}{11}$

(iv) $\frac{3}{7}\times \left(\frac{-2}{5}\right)$

(v) $\frac{3}{11}\times \frac{2}{5}$

(vi) $\frac{3}{-5}\times \frac{-5}{3}$

Sol. (i) $\frac{9}{2}\times \left(\frac{-7}{4}\right)=\frac{9\times (-7)}{2\times 4}=\frac{-63}{8}$

(ii) $\frac{3}{10}\times (-9)=\frac{3}{10}\times \frac{(-9)}{1}=\frac{3\times (-9)}{10\times 1}=\frac{-27}{10}$

(iii) $\frac{-6}{5}\times \frac{9}{11}=\frac{-6\times 9}{5\times 11}=\frac{-54}{55}$

(iv) $\frac{3}{7}\times \left(\frac{-2}{5}\right)=\frac{3\times (-2)}{7\times 5}=\frac{-6}{35}$

(v) $\frac{3}{11}\times \frac{2}{5}=\frac{3\times 2}{11\times 5}=\frac{6}{55}$

(vi) $\frac{3}{5}\times \frac{-5}{3}=\frac{3\times (-5)}{(-5)\times 3}=\frac{-15}{-15}=1$

P4. Find the value of:

(i) $(-4)÷\frac{2}{3}$

(ii) $\frac{-3}{5}÷2$

(iii) $\frac{-4}{5}÷(-3)$

(iv) $\frac{-1}{8}÷\frac{3}{4}$

(v) $\frac{-2}{13}÷\frac{1}{7}$

(vi) $\frac{-7}{12}÷\left(\frac{-2}{13}\right)$

(vii) $\frac{3}{13}÷\left(\frac{-4}{65}\right)$

Sol. (i) $\frac{-3}{5}÷2=-4\times \frac{3}{2}=\frac{-12}{2}=-6$

(ii) $\frac{-3}{5}÷2=\frac{-3}{5}\times \frac{1}{2}=\frac{-3\times 1}{5\times 2}=\frac{-3}{10}$

(iii) $\frac{-4}{5}÷(-3)=\frac{-4}{5}\times \frac{1}{-3}=\frac{(-4)\times 1}{5\times (-3)}$ =$\frac{-4}{-15}=\frac{4}{15}$

(iv) $\frac{-1}{-8}÷\frac{3}{4}=\frac{-1}{8}\times \frac{4}{3}=\frac{-1\times 4}{8\times 3}=\frac{-4}{24}=-\frac{1}{6}$

(v) $\frac{-2}{13}÷\frac{1}{7}=\frac{-2}{13}\times 7=\frac{-14}{13}$

(vi) $\frac{-7}{12}÷\left(\frac{-2}{13}\right)=\frac{-7}{12}\times \frac{13}{-2}=\frac{(-7)\times 13}{12\times (-2)}$ = $\frac{-91}{-24}=\frac{91}{24}$

(vii) $\frac{3}{4}÷\left(\frac{-4}{65}\right)=\frac{3}{13}\times \frac{65}{-4}=\frac{3\times 65}{13\times (-4)}$ = $\frac{195}{-52}=-\frac{15}{4}$