Fractions — Exercise 4(C)

Point out the proper and improper fractions from the following :

(i) $\dfrac{7}{9}$

(ii) $\dfrac{16}{11}$

(iii) $\dfrac{18}{25}$

(iv) $\dfrac{10}{10}$

(v) $\dfrac{37}{23}$

(vi) $\dfrac{21}{8}$

(vii) $\dfrac{56}{57}$

(viii) $\dfrac{137}{105}$

(ix) $\dfrac{2}{1}$

(x) $\dfrac{100}{101}$

Answer

(i) $\dfrac{7}{9}$

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).

Here, 7 < 9

Hence, $\dfrac{7}{9}$ is a proper fraction.

(ii) $\dfrac{16}{11}$

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Here, 16 > 11

Hence, $\dfrac{16}{11}$ is an improper fraction.

(iii) $\dfrac{18}{25}$

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).

Here, 18 < 25

Hence, $\dfrac{18}{25}$ is a proper fraction.

(iv) $\dfrac{10}{10}$

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Here, 10 = 10

Hence, $\dfrac{10}{10}$ is an improper fraction.

(v) $\dfrac{37}{23}$

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Here, 37 > 23

Hence, $\dfrac{37}{23}$ is an improper fraction.

(vi) $\dfrac{21}{8}$

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Here, 21 > 8

Hence, $\dfrac{21}{8}$ is an improper fraction.

(vii) $\dfrac{56}{57}$

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).

Here, 56 < 57

Hence, $\dfrac{56}{57}$ is a proper fraction.

(viii) $\dfrac{137}{105}$

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Here, 137 > 105

Hence, $\dfrac{137}{105}$ is an improper fraction.

(ix) $\dfrac{2}{1}$

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Here, 2 > 1

Hence, $\dfrac{2}{1}$ is an improper fraction.

(x) $\dfrac{100}{101}$

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).

Here, 100 < 101

Hence, $\dfrac{100}{101}$ is a proper fraction.

Convert each of the following mixed fractions into an improper fraction :

(i) $8\dfrac{3}{4}$

(ii) $5\dfrac{11}{13}$

(iii) $10\dfrac{7}{9}$

(iv) $33\dfrac{1}{3}$

(v) $9\dfrac{7}{16}$

Answer

(i) $8\dfrac{3}{4}$

$8\dfrac{3}{4} = \dfrac{8 \times 4 + 3}{4} = \dfrac{35}{4}$

Hence, the improper fraction = $\dfrac{35}{4}$.

(ii) $5\dfrac{11}{13}$

$5\dfrac{11}{13} = \dfrac{5 \times 13 + 11}{13} = \dfrac{76}{13}$

Hence, the improper fraction = $\dfrac{76}{13}$.

(iii) $10\dfrac{7}{9}$

$10\dfrac{7}{9} = \dfrac{10 \times 9 + 7}{9} = \dfrac{97}{9}$

Hence, the improper fraction = $\dfrac{97}{9}$.

(iv) $33\dfrac{1}{3}$

$33\dfrac{1}{3} = \dfrac{33 \times 3 + 1}{3} = \dfrac{100}{3}$

Hence, the improper fraction = $\dfrac{100}{3}$.

(v) $9\dfrac{7}{16}$

$9\dfrac{7}{16} = \dfrac{9 \times 16 + 7}{16} = \dfrac{151}{16}$

Hence, the improper fraction = $\dfrac{151}{16}$.

Convert each of the following improper fractions into a mixed fraction :

(i) $\dfrac{31}{5}$

(ii) $\dfrac{80}{7}$

(iii) $\dfrac{107}{3}$

(iv) $\dfrac{115}{13}$

(v) $\dfrac{200}{9}$

Answer

(i) $\dfrac{31}{5}$

On dividing 31 by 5, we get quotient = 6 and remainder = 1.

$\therefore \dfrac{31}{5} = 6 + \dfrac{1}{5} = 6\dfrac{1}{5}$.

Hence, the mixed fraction is $6\dfrac{1}{5}$.

(ii) $\dfrac{80}{7}$

On dividing 80 by 7, we get quotient = 11 and remainder = 3.

$\therefore \dfrac{80}{7} = 11 + \dfrac{3}{7} = 11\dfrac{3}{7}$.

Hence, the mixed fraction is $11\dfrac{3}{7}$.

(iii) $\dfrac{107}{3}$

On dividing 107 by 3, we get quotient = 35 and remainder = 2.

$\therefore \dfrac{107}{3} = 35 + \dfrac{2}{3} = 35\dfrac{2}{3}$.

Hence, the mixed fraction is $35\dfrac{2}{3}$.

(iv) $\dfrac{115}{13}$

On dividing 115 by 13, we get quotient = 8 and remainder = 11.

$\therefore \dfrac{115}{13} = 8 + \dfrac{11}{13} = 8\dfrac{11}{13}$.

Hence, the mixed fraction is $8\dfrac{11}{13}$.

(v) $\dfrac{200}{9}$

On dividing 200 by 9, we get quotient = 22 and remainder = 2.

$\therefore \dfrac{200}{9} = 22 + \dfrac{2}{9} = 22\dfrac{2}{9}.$

Hence, the mixed fraction is $22\dfrac{2}{9}$.

Convert each of the following sets of unlike fractions into that of like fractions :

(i) $\dfrac{4}{5}, \dfrac{7}{10}, \dfrac{11}{15}, \dfrac{13}{20}$

(ii) $\dfrac{2}{3}, \dfrac{1}{4}, \dfrac{5}{6}, \dfrac{7}{8}, \dfrac{11}{12}$

(iii) $\dfrac{1}{3}, \dfrac{3}{4}, \dfrac{5}{12}, \dfrac{9}{16}, \dfrac{17}{24}$

(iv) $\dfrac{2}{3}, \dfrac{1}{6}, \dfrac{5}{9}, \dfrac{7}{12}, \dfrac{13}{18}$

(v) $\dfrac{1}{2}, \dfrac{4}{7}, \dfrac{9}{14}, \dfrac{11}{21}, \dfrac{37}{42}$

(vi) $\dfrac{2}{7}, \dfrac{5}{8}, \dfrac{11}{14}, \dfrac{9}{16}, \dfrac{3}{4}$

Answer

(i) $\dfrac{4}{5}, \dfrac{7}{10}, \dfrac{11}{15}, \dfrac{13}{20}$

Denominators: 5, 10, 15, 20

LCM = 60.

Equivalent fractions:

$\Rightarrow \dfrac{4 \times 12}{5 \times 12}, \dfrac{7 \times 6}{10 \times 6}, \dfrac{11 \times 4}{15 \times 4}, \dfrac{13 \times 3}{20 \times 3}\\[1em] \Rightarrow \dfrac{48}{60}, \dfrac{42}{60}, \dfrac{44}{60}, \dfrac{39}{60}$

Hence, the like fractions are $\dfrac{48}{60}, \dfrac{42}{60}, \dfrac{44}{60}, \dfrac{39}{60}$.

(ii) $\dfrac{2}{3}, \dfrac{1}{4}, \dfrac{5}{6}, \dfrac{7}{8}, \dfrac{11}{12}$

Denominators: 3, 4, 6, 8 and 12

LCM = 24.

Equivalent fractions:

$\Rightarrow \dfrac{2 \times 8}{3 \times 8}, \dfrac{1 \times 6}{4 \times 6}, \dfrac{5 \times 4}{6 \times 4}, \dfrac{7 \times 3}{8 \times 3}, \dfrac{11 \times 2}{12 \times 2}\\[1em] \Rightarrow \dfrac{16}{24}, \dfrac{6}{24}, \dfrac{20}{24}, \dfrac{21}{24}, \dfrac{22}{24}$

Hence, the like fractions are $\dfrac{16}{24}, \dfrac{6}{24}, \dfrac{20}{24}, \dfrac{21}{24}, \dfrac{22}{24}$.

(iii) $\dfrac{1}{3}, \dfrac{3}{4}, \dfrac{5}{12}, \dfrac{9}{16}, \dfrac{17}{24}$

Denominators: 3, 4, 12, 16, 24

LCM = 48.

Equivalent fractions:

$\Rightarrow \dfrac{1 \times 16}{3 \times 16}, \dfrac{3 \times 12}{4 \times 12}, \dfrac{5 \times 4}{12 \times 4}, \dfrac{9 \times 3}{16 \times 3}, \dfrac{17 \times 2}{24 \times 2}\\[1em] \Rightarrow \dfrac{16}{48}, \dfrac{36}{48}, \dfrac{20}{48}, \dfrac{27}{48}, \dfrac{34}{48}$

Hence, the like fractions are $\dfrac{16}{48}, \dfrac{36}{48}, \dfrac{20}{48}, \dfrac{27}{48}, \dfrac{34}{48}$.

(iv) $\dfrac{2}{3}, \dfrac{1}{6}, \dfrac{5}{9}, \dfrac{7}{12}, \dfrac{13}{18}$

Denominators: 3, 6, 9, 12, 18

LCM = 36.

Equivalent fractions:

$\Rightarrow \dfrac{2 \times 12}{3 \times 12}, \dfrac{1 \times 6}{6 \times 6}, \dfrac{5 \times 4}{9 \times 4}, \dfrac{7 \times 3}{12 \times 3}, \dfrac{13 \times 2}{18 \times 2}\\[1em] \Rightarrow \dfrac{24}{36}, \dfrac{6}{36}, \dfrac{20}{36}, \dfrac{21}{36}, \dfrac{26}{36}$

Hence, the like fractions are $\dfrac{24}{36}, \dfrac{6}{36}, \dfrac{20}{36}, \dfrac{21}{36}, \dfrac{26}{36}$.

(v) $\dfrac{1}{2}, \dfrac{4}{7}, \dfrac{9}{14}, \dfrac{11}{21}, \dfrac{37}{42}$

Denominators: 2, 7, 14, 21, 42

LCM = 42.

Equivalent fractions:

$\Rightarrow \dfrac{1 \times 21}{2 \times 21}, \dfrac{4 \times 6}{7 \times 6}, \dfrac{9 \times 3}{14 \times 3}, \dfrac{11 \times 2}{21 \times 2}, \dfrac{37 \times 1}{42 \times 1}\\[1em] \Rightarrow \dfrac{21}{42}, \dfrac{24}{42}, \dfrac{27}{42}, \dfrac{22}{42}, \dfrac{37}{42}$

Hence, the like fractions are $\dfrac{21}{42}, \dfrac{24}{42}, \dfrac{27}{42}, \dfrac{22}{42}, \dfrac{37}{42}$.

(vi) $\dfrac{2}{7}, \dfrac{5}{8}, \dfrac{11}{14}, \dfrac{9}{16}, \dfrac{3}{4}$

Denominators: 7, 8, 14, 16, 4

LCM = 112.

Equivalent fractions:

$\Rightarrow \dfrac{2 \times 16}{7 \times 16}, \dfrac{5 \times 14}{8 \times 14}, \dfrac{11 \times 8}{14 \times 8}, \dfrac{9 \times 7}{16 \times 7}, \dfrac{3 \times 28}{4 \times 28}\\[1em] \Rightarrow \dfrac{32}{112}, \dfrac{70}{112}, \dfrac{88}{112}, \dfrac{63}{112}, \dfrac{84}{112}$

Hence, the like fractions are $\dfrac{32}{112}, \dfrac{70}{112}, \dfrac{88}{112}, \dfrac{63}{112}, \dfrac{84}{112}$.

Fill in the placeholders with > or < :

(i) $\dfrac{7}{9} {\boxed{\phantom{63}}} \dfrac{5}{9}$

(ii) $\dfrac{9}{13} {\boxed{\phantom{63}}} \dfrac{11}{13}$

(iii) $\dfrac{3}{5} {\boxed{\phantom{63}}} \dfrac{3}{4}$

(iv) $\dfrac{7}{9} {\boxed{\phantom{63}}} \dfrac{7}{11}$

(v) $\dfrac{5}{8} {\boxed{\phantom{63}}} \dfrac{5}{6}$

(vi) $\dfrac{7}{9} {\boxed{\phantom{63}}} \dfrac{6}{11}$

(vii) $\dfrac{6}{5} {\boxed{\phantom{63}}} \dfrac{5}{4}$

(viii) $\dfrac{7}{11} {\boxed{\phantom{63}}} \dfrac{8}{13}$

(ix) $\dfrac{10}{13} {\boxed{\phantom{63}}} \dfrac{13}{16}$

(x) $\dfrac{2}{9} {\boxed{\phantom{63}}} \dfrac{3}{14}$

(xi) $\dfrac{7}{12} {\boxed{\phantom{63}}} \dfrac{5}{9}$

(xii) $\dfrac{15}{19} {\boxed{\phantom{63}}} \dfrac{3}{4}$

Answer

(i) $\dfrac{7}{9} {\boxed{\phantom{63}}} \dfrac{5}{9}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

Hence, $\dfrac{7}{9} {\boxed{\gt}} \dfrac{5}{9}$.

(ii) $\dfrac{9}{13} {\boxed{\phantom{63}}} \dfrac{11}{13}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

Hence, $\dfrac{9}{13} {\boxed \lt} \dfrac{11}{13}$.

(iii) $\dfrac{3}{5} {\boxed{\phantom{63}}} \dfrac{3}{4}$

Among two fractions with same numerator, the one with smaller denominator is greater of the two.

Hence, $\dfrac{3}{5} {\boxed{\lt}} \dfrac{3}{4}$.

(iv) $\dfrac{7}{9}$ ${\boxed{\phantom{63}}}$ $\dfrac{7}{11}$

Among two fractions with same numerator, the one with smaller denominator is greater of the two.

Hence, $\dfrac{7}{9}$ ${\boxed{\gt}}$ $\dfrac{7}{11}$

(v) $\dfrac{5}{8} {\boxed{\phantom{63}}} \dfrac{5}{6}$

Among two fractions with same numerator, the one with smaller denominator is greater of the two.

Hence, $\dfrac{5}{8} {\boxed{\lt}} \dfrac{5}{6}$.

(vi) $\dfrac{7}{9} {\boxed{\phantom{63}}} \dfrac{6}{11}$

Cross multiply; 7 x 11 and 6 x 9

⇒ 77 and 54

Now, comparing both numbers;

⇒ 77 > 54

Hence, $\dfrac{7}{9} {\boxed{\gt}} \dfrac{6}{11}$.

(vii) $\dfrac{6}{5} {\boxed{\phantom{63}}} \dfrac{5}{4}$

Cross multiply; 6 x 4 and 5 x 5

⇒ 24 and 25

Now, comparing both numbers;

⇒ 24 < 25

Hence, $\dfrac{6}{5} {\boxed{\lt}} \dfrac{5}{4}$.

(viii) $\dfrac{7}{11} {\boxed{\phantom{63}}} \dfrac{8}{13}$

Cross multiply; 7 x 13 and 8 x 11

⇒ 91 and 88

Now, comparing both numbers;

⇒ 91 > 88

Hence, $\dfrac{7}{11} {\boxed{\gt}} \dfrac{8}{13}$.

(ix) $\dfrac{10}{13} {\boxed{\phantom{63}}} \dfrac{13}{16}$

Cross multiply; 10 x 16 and 13 x 13

⇒ 160 and 169

Now, comparing both numbers;

⇒ 160 < 169

Hence, $\dfrac{10}{13} {\boxed{\lt}} \dfrac{13}{16}$.

(x) $\dfrac{2}{9} {\boxed{\phantom{63}}} \dfrac{3}{14}$

Cross multiply; 2 x 14 and 3 x 9

⇒ 28 and 27

Now, comparing both numbers;

⇒ 28 > 27

Hence, $\dfrac{2}{9} {\boxed{\gt}} \dfrac{3}{14}$.

(xi) $\dfrac{7}{12} {\boxed{\phantom{63}}} \dfrac{5}{9}$

Cross multiply; 7 x 9 and 12 x 5

⇒ 63 and 60

Now, comparing both numbers;

⇒ 63 > 60

Hence, $\dfrac{7}{12} {\boxed{\gt}} \dfrac{5}{9}$.

(xii) $\dfrac{15}{19} {\boxed{\phantom{63}}} \dfrac{3}{4}$

Cross multiply; 15 x 4 and 3 x 19

⇒ 60 and 57

Now, comparing both numbers;

⇒ 60 > 57

Hence, $\dfrac{15}{19} {\boxed{\gt}} \dfrac{3}{4}$.

Arrange the following fractions in ascending order :

(i) $\dfrac{3}{11}, \dfrac{9}{11}, \dfrac{4}{11}, \dfrac{5}{11}, \dfrac{1}{11}$

(ii) $\dfrac{2}{13}, \dfrac{2}{9}, \dfrac{2}{15}, \dfrac{2}{7}, \dfrac{2}{5}$

(iii) $\dfrac{2}{3}, \dfrac{5}{6}, \dfrac{7}{9}, \dfrac{11}{12}, \dfrac{13}{18}$

(iv) $\dfrac{2}{3}, \dfrac{1}{4}, \dfrac{5}{6}, \dfrac{3}{8}, \dfrac{7}{12}$

Answer

(i) $\dfrac{3}{11}, \dfrac{9}{11}, \dfrac{4}{11}, \dfrac{5}{11}, \dfrac{1}{11}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

Hence, $\dfrac{1}{11} \lt \dfrac{3}{11} \lt \dfrac{4}{11} \lt \dfrac{5}{11} \lt \dfrac{9}{11}$

(ii) $\dfrac{2}{13}, \dfrac{2}{9}, \dfrac{2}{15}, \dfrac{2}{7}, \dfrac{2}{5}$

Among two fractions with same numerator, the one with smaller denominator is greater of the two.

Hence, $\dfrac{2}{15} \lt \dfrac{2}{13} \lt \dfrac{2}{9} \lt \dfrac{2}{7} \lt \dfrac{2}{5}$

(iii) $\dfrac{2}{3}, \dfrac{5}{6}, \dfrac{7}{9}, \dfrac{11}{12}, \dfrac{13}{18}$

LCM of the denominators (3, 6, 9, 12, 18) = 36

$\Rightarrow \dfrac{2 \times 12}{3 \times 12}, \dfrac{5 \times 6}{6 \times 6}, \dfrac{7 \times 4}{9 \times 4}, \dfrac{11 \times 3}{12 \times 3}, \dfrac{13 \times 2}{18 \times 2}\\[1em] \Rightarrow \dfrac{24}{36}, \dfrac{30}{36}, \dfrac{28}{36}, \dfrac{33}{36}, \dfrac{26}{36}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{24}{36} \lt \dfrac{26}{36} \lt \dfrac{28}{36} \lt \dfrac{30}{36} \lt \dfrac{33}{36}$

Hence, $\dfrac{2}{3} \lt \dfrac{13}{18} \lt \dfrac{7}{9} \lt \dfrac{5}{6} \lt \dfrac{11}{12}$

(iv) $\dfrac{2}{3}, \dfrac{1}{4}, \dfrac{5}{6}, \dfrac{3}{8}, \dfrac{7}{12}$

LCM of the denominators (3, 4, 6, 8, 12) = 24

$\Rightarrow \dfrac{2 \times 8}{3 \times 8}, \dfrac{1 \times 6}{4 \times 6}, \dfrac{5 \times 4}{6 \times 4}, \dfrac{3 \times 3}{8 \times 3}, \dfrac{7 \times 2}{12 \times 2}\\[1em] \Rightarrow \dfrac{16}{24}, \dfrac{6}{24}, \dfrac{20}{24}, \dfrac{9}{24}, \dfrac{14}{24}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{6}{24} \lt \dfrac{9}{24} \lt \dfrac{14}{24} \lt \dfrac{16}{24} \lt \dfrac{20}{24}$

Hence, $\dfrac{1}{4} \lt \dfrac{3}{8} \lt \dfrac{7}{12} \lt \dfrac{2}{3} \lt \dfrac{5}{6}$

Arrange the following fractions in descending order :

(i) $\dfrac{5}{12}, \dfrac{1}{12}, \dfrac{7}{12}, \dfrac{11}{12}, \dfrac{9}{12}$

(ii) $\dfrac{4}{7}, \dfrac{4}{3}, \dfrac{4}{9}, \dfrac{4}{5}, \dfrac{4}{11}$

(iii) $\dfrac{2}{3}, \dfrac{5}{6}, \dfrac{7}{9}, \dfrac{3}{4}, \dfrac{1}{2}$

(iv) $\dfrac{2}{3}, \dfrac{3}{5}, \dfrac{7}{10}, \dfrac{8}{15}, \dfrac{11}{20}$

(v) $\dfrac{17}{32}, \dfrac{7}{12}, \dfrac{19}{48}, \dfrac{13}{24}, \dfrac{9}{16}$

(vi) $\dfrac{5}{6}, \dfrac{7}{9}, \dfrac{17}{24}, \dfrac{3}{4}, \dfrac{23}{36}$

Answer

(i) $\dfrac{5}{12}, \dfrac{1}{12}, \dfrac{7}{12}, \dfrac{11}{12}, \dfrac{9}{12}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{11}{12} \gt \dfrac{9}{12} \gt \dfrac{7}{12} \gt \dfrac{5}{12} \gt \dfrac{1}{12}$

Hence, $\dfrac{11}{12} \gt \dfrac{9}{12} \gt \dfrac{7}{12} \gt \dfrac{5}{12} \gt \dfrac{1}{12}$.

(ii) $\dfrac{4}{7}, \dfrac{4}{3}, \dfrac{4}{9}, \dfrac{4}{5}, \dfrac{4}{11}$

Among two fractions with same numerator, the one with smaller denominator is greater of the two.

$\dfrac{4}{3} \gt \dfrac{4}{5} \gt \dfrac{4}{7} \gt \dfrac{4}{9} \gt \dfrac{4}{11}$

Hence, $\dfrac{4}{3} \gt \dfrac{4}{5} \gt \dfrac{4}{7} \gt \dfrac{4}{9} \gt \dfrac{4}{11}$.

(iii) $\dfrac{2}{3}, \dfrac{5}{6}, \dfrac{7}{9}, \dfrac{3}{4}, \dfrac{1}{2}$

LCM of the denominators (3, 6, 9, 4, 2) = 36

$\Rightarrow \dfrac{2 \times 12}{3 \times 12}, \dfrac{5 \times 6}{6 \times 6}, \dfrac{7 \times 4}{9 \times 4}, \dfrac{3 \times 9}{4 \times 9}, \dfrac{1 \times 18}{2 \times 18}\\[1em] \Rightarrow \dfrac{24}{36}, \dfrac{30}{36}, \dfrac{28}{36}, \dfrac{27}{36}, \dfrac{18}{36}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{30}{36} \gt \dfrac{28}{36} \gt \dfrac{27}{36} \gt \dfrac{24}{36} \gt \dfrac{18}{36}$

Hence, $\dfrac{5}{6} \gt \dfrac{7}{9} \gt \dfrac{3}{4} \gt \dfrac{2}{3} \gt \dfrac{1}{2}$

(iv) $\dfrac{2}{3}, \dfrac{3}{5}, \dfrac{7}{10}, \dfrac{8}{15}, \dfrac{11}{20}$

LCM of the denominators (3, 5, 10, 15, 20) = 60

$\Rightarrow \dfrac{2 \times 20}{3 \times 20}, \dfrac{3 \times 12}{5 \times 12}, \dfrac{7 \times 6}{10 \times 6}, \dfrac{8 \times 4}{15 \times 4}, \dfrac{11 \times 3}{20 \times 3}\\[1em] \Rightarrow \dfrac{40}{60}, \dfrac{36}{60}, \dfrac{42}{60}, \dfrac{32}{60}, \dfrac{33}{60}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{42}{60} \gt \dfrac{40}{60} \gt \dfrac{36}{60} \gt \dfrac{33}{60} \gt \dfrac{32}{60}$

Hence, $\dfrac{7}{10} \gt \dfrac{2}{3} \gt \dfrac{3}{5} \gt \dfrac{11}{20} \gt \dfrac{8}{15}$

(v) $\dfrac{17}{32}, \dfrac{7}{12}, \dfrac{19}{48}, \dfrac{13}{24}, \dfrac{9}{16}$

LCM of the denominators (32, 12, 48, 24, 16) = 96

$\Rightarrow \dfrac{17 \times 3}{32 \times 3}, \dfrac{7 \times 8}{12 \times 8}, \dfrac{19 \times 2}{48 \times 2}, \dfrac{13 \times 4}{24 \times 4}, \dfrac{9 \times 6}{16 \times 6}\\[1em] \Rightarrow \dfrac{51}{96}, \dfrac{56}{96}, \dfrac{38}{96}, \dfrac{52}{96}, \dfrac{54}{96}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{56}{96} \gt \dfrac{54}{96} \gt \dfrac{52}{96} \gt \dfrac{51}{96} \gt \dfrac{38}{96}$

Hence, $\dfrac{7}{12} \gt \dfrac{9}{16} \gt \dfrac{13}{24} \gt \dfrac{17}{32} \gt \dfrac{19}{48}$

(vi) $\dfrac{5}{6}, \dfrac{7}{9}, \dfrac{17}{24}, \dfrac{3}{4}, \dfrac{23}{36}$

LCM of the denominators (6, 9, 24, 4, 36) = 72

$\Rightarrow \dfrac{5 \times 12}{6 \times 12}, \dfrac{7 \times 8}{9 \times 8}, \dfrac{17 \times 3}{24 \times 3}, \dfrac{3 \times 18}{4 \times 18}, \dfrac{23 \times 2}{36 \times 2}\\[1em] \Rightarrow \dfrac{60}{72}, \dfrac{56}{72}, \dfrac{51}{72}, \dfrac{54}{72}, \dfrac{46}{72}$

Among two fractions with same denominator, the one with greater numerator is greater of the two.

$\dfrac{60}{72} \gt \dfrac{56}{72} \gt \dfrac{54}{72} \gt \dfrac{51}{72} \gt \dfrac{46}{72}$

Hence, $\dfrac{5}{6} \gt \dfrac{7}{9} \gt \dfrac{3}{4} \gt \dfrac{17}{24} \gt \dfrac{23}{36}$