Fractions — Exercise 4(A)

What fractions do the shaded parts in each of the following figures represent?

Answer

(i) Total parts = 8

Shaded parts = 5

Fraction = $\dfrac{\text{Shaded parts}}{\text{Total parts}}$

= $\dfrac{5}{8}$.

Hence, fraction = $\dfrac{5}{8}$.

(ii) Total parts = 12

Shaded parts = 7

Fraction = $\dfrac{\text{Shaded parts}}{\text{Total parts}}$

= $\dfrac{7}{12}$.

Hence, fraction = $\dfrac{7}{12}$.

(iii) Total parts = 7

Shaded parts = 2

Fraction = $\dfrac{\text{Shaded parts}}{\text{Total parts}}$

= $\dfrac{2}{7}$

Hence, fraction = $\dfrac{2}{7}$.

Write a fraction for each of the following :

(i) 4 parts out of 9 equal parts

(ii) 5 parts out of 11 equal parts

(iii) two-fifths

(iv) four-sevenths

(v) seven-tenths

(vi) six-eighths

Answer

(i) Total parts = 9

Given parts = 4

Fraction = $\dfrac{\text{Given parts}}{\text{Total parts}}$

= $\dfrac{4}{9}$

Hence, fraction = $\dfrac{4}{9}$.

(ii) Total parts = 11

Given parts = 5

Fraction = $\dfrac{\text{Given parts}}{\text{Total parts}}$

= $\dfrac{5}{11}$

Hence, fraction = $\dfrac{5}{11}$.

(iii) Total parts = 5

Given parts = 2

Fraction = $\dfrac{\text{Given parts}}{\text{Total parts}}$

= $\dfrac{2}{5}$

Hence, fraction = $\dfrac{2}{5}$.

(iv) Total parts = 7

Given parts = 4

Fraction = $\dfrac{\text{Given parts}}{\text{Total parts}}$

= $\dfrac{4}{7}$

Hence, fraction = $\dfrac{4}{7}$.

(v) Total parts = 10

Given parts = 7

Fraction = $\dfrac{\text{Given parts}}{\text{Total parts}}$

= $\dfrac{7}{10}$

Hence, fraction = $\dfrac{7}{10}$.

(vi) Total parts = 8

Given parts = 6

Fraction = $\dfrac{\text{Given parts}}{\text{Total parts}}$

= $\dfrac{6}{8}$

Hence, fraction = $\dfrac{6}{8}$.

Point out the numerator and denominator in each of the following fractions :

(i) $\dfrac{5}{6}$

(ii) $\dfrac{3}{7}$

(iii) $\dfrac{9}{14}$

(iv) $\dfrac{1}{20}$

(v) $\dfrac{12}{25}$

Answer

(i) $\dfrac{5}{6}$

Hence, numerator = 5 and denominator = 6.

(ii) $\dfrac{3}{7}$

Hence, numerator = 3 and denominator = 7.

(iii) $\dfrac{9}{14}$

Hence, numerator = 9 and denominator = 14.

(iv) $\dfrac{1}{20}$

Hence, numerator = 1 and denominator = 20.

(v) $\dfrac{12}{25}$

Hence, numerator = 12 and denominator = 25.

Write five fractions equivalent to each of the following :

(i) $\dfrac{2}{3}$

(ii) $\dfrac{3}{4}$

(iii) $\dfrac{5}{6}$

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

(v) $\dfrac{3}{14}$

Answer

(i) $\dfrac{2}{3}$

To find equivalent fractions, we multiply the numerator and denominator by a sequence of numbers (e.g., 2, 3, 4, 5, 6).

Multiply by 2: $\dfrac{2 \times 2}{3 \times 2} = \dfrac{4}{6}$

Multiply by 3: $\dfrac{2 \times 3}{3 \times 3} = \dfrac{6}{9}$

Multiply by 4: $\dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}$

Multiply by 5: $\dfrac{2 \times 5}{3 \times 5} = \dfrac{10}{15}$

Multiply by 6: $\dfrac{2 \times 6}{3 \times 6} = \dfrac{12}{18}$

Hence, the equivalent fractions are : $\dfrac{4}{6}, \dfrac{6}{9}, \dfrac{8}{12}, \dfrac{10}{15}, \dfrac{12}{18}$.

(ii) $\dfrac{3}{4}$

To find equivalent fractions, we multiply the numerator and denominator by a sequence of numbers (e.g., 2, 3, 4, 5, 6).

Multiply by 2: $\dfrac{3 \times 2}{4 \times 2} = \dfrac{6}{8}$

Multiply by 3: $\dfrac{3 \times 3}{4 \times 3} = \dfrac{9}{12}$

Multiply by 4: $\dfrac{3 \times 4}{4 \times 4} = \dfrac{12}{16}$

Multiply by 5: $\dfrac{3 \times 5}{4 \times 5} = \dfrac{15}{20}$

Multiply by 6: $\dfrac{3 \times 6}{4 \times 6} = \dfrac{18}{24}$

Hence, the equivalent fractions are: $\dfrac{6}{8}, \dfrac{9}{12}, \dfrac{12}{16}, \dfrac{15}{20}, \dfrac{18}{24}$.

(iii) $\dfrac{5}{6}$

To find equivalent fractions, we multiply the numerator and denominator by a sequence of numbers (e.g., 2, 3, 4, 5, 6).

Multiply by 2: $\dfrac{5 \times 2}{6 \times 2} = \dfrac{10}{12}$

Multiply by 3: $\dfrac{5 \times 3}{6 \times 3} = \dfrac{15}{18}$

Multiply by 4: $\dfrac{5 \times 4}{6 \times 4} = \dfrac{20}{24}$

Multiply by 5: $\dfrac{5 \times 5}{6 \times 5} = \dfrac{25}{30}$

Multiply by 6: $\dfrac{5 \times 6}{6 \times 6} = \dfrac{30}{36}$

Hence, the equivalent fractions are: $\dfrac{10}{12}, \dfrac{15}{18}, \dfrac{20}{24}, \dfrac{25}{30}, \dfrac{30}{36}$.

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

To find equivalent fractions, we multiply the numerator and denominator by a sequence of numbers (e.g., 2, 3, 4, 5, 6).

Multiply by 2: $\dfrac{7 \times 2}{9 \times 2} = \dfrac{14}{18}$

Multiply by 3: $\dfrac{7 \times 3}{9 \times 3} = \dfrac{21}{27}$

Multiply by 4: $\dfrac{7 \times 4}{9 \times 4} = \dfrac{28}{36}$

Multiply by 5: $\dfrac{7 \times 5}{9 \times 5} = \dfrac{35}{45}$

Multiply by 6: $\dfrac{7 \times 6}{9 \times 6} = \dfrac{42}{54}$

Hence, the equivalent fractions are: $\dfrac{14}{18}, \dfrac{21}{27}, \dfrac{28}{36}, \dfrac{35}{45}, \dfrac{42}{54}$.

(v) $\dfrac{3}{14}$

To find equivalent fractions, we multiply the numerator and denominator by a sequence of numbers (e.g., 2, 3, 4, 5, 6).

Multiply by 2: $\dfrac{3 \times 2}{14 \times 2} = \dfrac{6}{28}$

Multiply by 3: $\dfrac{3 \times 3}{14 \times 3} = \dfrac{9}{42}$

Multiply by 4: $\dfrac{3 \times 4}{14 \times 4} = \dfrac{12}{56}$

Multiply by 5: $\dfrac{3 \times 5}{14 \times 5} = \dfrac{15}{70}$

Multiply by 6: $\dfrac{3 \times 6}{14 \times 6} = \dfrac{18}{84}$

Hence, the equivalent fractions are: $\dfrac{6}{28}, \dfrac{9}{42}, \dfrac{12}{56}, \dfrac{15}{70}, \dfrac{18}{84}$.

Which of the following are the pairs of equivalent fractions?

(i) $\dfrac{4}{5}$ and $\dfrac{24}{30}$

(ii) $\dfrac{6}{8}$ and $\dfrac{18}{24}$

(iii) $\dfrac{9}{16}$ and $\dfrac{3}{4}$

(iv) $\dfrac{8}{12}$ and $\dfrac{14}{21}$

(v) $\dfrac{7}{11}$ and $\dfrac{21}{32}$

(vi) $\dfrac{5}{12}$ and $\dfrac{35}{84}$

Answer

(i) $\dfrac{4}{5}$ and $\dfrac{24}{30}$

By cross multiplication,

So, 4 x 30 = 120 and 5 x 24 = 120

Here, the cross products are equal.

Hence, $\dfrac{4}{5} \text{ and } \dfrac{24}{30}$ are equivalent fractions.

(ii) $\dfrac{6}{8}$ and $\dfrac{18}{24}$

By cross multiplication,

So, 6 x 24 = 144 and 8 x 18 = 144

Here, the cross products are equal.

Hence, $\dfrac{6}{8}\text{ and } \dfrac{18}{24}$ are equivalent fractions.

(iii) $\dfrac{9}{16}$ and $\dfrac{3}{4}$

By cross multiplication,

So, 9 x 4 = 36 and 16 x 3 = 48

Here, the cross products are not equal.

Hence, $\dfrac{9}{16}\text{ and } \dfrac{3}{4}$ are not equivalent fractions.

(iv) $\dfrac{8}{12}$ and $\dfrac{14}{21}$

By cross multiplication,

So, 8 x 21 = 168 and 12 x 14 = 168

Here, the cross products are equal.

Hence, $\dfrac{8}{12}\text{ and } \dfrac{14}{21}$ are equivalent fractions.

(v) $\dfrac{7}{11}$ and $\dfrac{21}{32}$

By cross multiplication,

So, 7 x 32 = 224 and 11 x 21 = 231

Here, the cross products are not equal.

Hence, $\dfrac{7}{11}\text{ and } \dfrac{21}{32}$ are not equivalent fractions.

(vi) $\dfrac{5}{12}$ and $\dfrac{35}{84}$

By cross multiplication,

So, 5 x 84 = 420 and 12 x 35 = 420

Here, the cross products are equal.

Hence, $\dfrac{5}{12}\text{ and } \dfrac{35}{84}$ are equivalent fractions.

Write an equivalent fraction of :

(i) $\dfrac{4}{5}$ with numerator 32

(ii) $\dfrac{7}{9}$ with numerator 42

(iii) $\dfrac{3}{7}$ with denominator 63

(iv) $\dfrac{10}{13}$ with denominator 78

Answer

(i) $\dfrac{4}{5}$ with numerator 32

To find an equivalent fraction whose numerator is 32, we multiply the fraction $\dfrac{4}{5}$ with 8,

$\dfrac{4}{5} = \dfrac{4 \times 8}{5 \times 8} = \dfrac{32}{40}$.

Hence, an equivalent fraction of $\dfrac{4}{5} = \dfrac{32}{40}$.

(ii) $\dfrac{7}{9}$ with numerator 42

To find an equivalent fraction whose numerator is 42, we multiply the fraction $\dfrac{7}{9}$ with 6,

$\dfrac{7}{9} = \dfrac{7 \times 6}{9 \times 6} = \dfrac{42}{54}$.

Hence, an equivalent fraction of $\dfrac{7}{9} = \dfrac{42}{54}$.

(iii) $\dfrac{3}{7}$ with denominator 63

To find an equivalent fraction whose denominator is 63, we multiply the fraction $\dfrac{3}{7}$ with 9,

$\dfrac{3}{7} = \dfrac{3 \times 9}{7 \times 9} = \dfrac{27}{63}$.

Hence, an equivalent fraction of $\dfrac{3}{7} = \dfrac{27}{63}$.

(iv) $\dfrac{10}{13}$ with denominator 78

To find an equivalent fraction whose denominator is 78, we multiply the fraction $\dfrac{10}{13}$ with 6,

$\dfrac{10}{13} = \dfrac{10 \times 6}{13 \times 6} = \dfrac{60}{78}$.

Hence, an equivalent fraction of $\dfrac{10}{13} = \dfrac{60}{78}$.

Write an equivalent fraction of :

(i) $\dfrac{32}{48}$ with numerator 2

(ii) $\dfrac{21}{28}$ with numerator 3

(iii) $\dfrac{24}{56}$ with denominator 7

(iv) $\dfrac{121}{132}$ with denominator 12

Answer

(i) $\dfrac{32}{48}$ with numerator 2

The H.C.F. of 32 and 48 is 16. Thus,

To find an equivalent fraction whose numerator is 2, we divide the numerator and denominator of the given fraction by 16,

$\dfrac{32}{48} = \dfrac{32 ÷ 16}{48 ÷ 16} = \dfrac{2}{3}$.

Hence, an equivalent fraction of $\dfrac{32}{48} = \dfrac{2}{3}$.

(ii) $\dfrac{21}{28}$ with numerator 3

The H.C.F. of 21 and 28 is 7. Thus,

To find an equivalent fraction whose numerator is 3, we divide the numerator and denominator of the given fraction by 7,

$\dfrac{21}{28} = \dfrac{21 ÷ 7}{28 ÷ 7} = \dfrac{3}{4}$.

Hence, an equivalent fraction of $\dfrac{21}{28} = \dfrac{3}{4}$.

(iii) $\dfrac{24}{56}$ with denominator 7

The H.C.F. of 24 and 56 is 8. Thus,

To find an equivalent fraction whose denominator is 7, we divide the numerator and denominator of the given fraction by 8,

$\dfrac{24}{56} = \dfrac{24 ÷ 8}{56 ÷ 8} = \dfrac{3}{7}$.

Hence, an equivalent fraction of $\dfrac{24}{56} = \dfrac{3}{7}$.

(iv) $\dfrac{121}{132}$ with denominator 12

The H.C.F. of 121 and 132 is 11. Thus,

To find an equivalent fraction whose denominator is 12, we divide the numerator and denominator of the given fraction by 11,

$\dfrac{121}{132} = \dfrac{121 ÷ 11}{132 ÷ 11} = \dfrac{11}{12}$.

Hence, an equivalent fraction of $\dfrac{121}{132} = \dfrac{11}{12}$.

Write an equivalent fraction of :

(i) $\dfrac{45}{54}$ with numerator 20.

(ii) $\dfrac{75}{90}$ with denominator 42.

Answer

(i) $\dfrac{45}{54}$ with numerator 20.

The H.C.F. of 45 and 54 is 9. Thus,

To find an equivalent fraction whose numerator is 20,

We divide the numerator and denominator of the given fraction by 9,

$\dfrac{45}{54} = \dfrac{45 ÷ 9}{54 ÷ 9} = \dfrac{5}{6}$.

Now, to get a fraction with numerator 20, we multiply both the numerator and denominator by 4,

$\dfrac{5}{6} = \dfrac{5 \times 4}{6 \times 4} = \dfrac{20}{24}$.

Hence, the equivalent fraction = $\dfrac{20}{24}$.

(ii) $\dfrac{75}{90}$ with denominator 42.

The H.C.F. of 75 and 90 is 15. Thus,

To find an equivalent fraction whose denominator is 42,

We divide the numerator and denominator of the given fraction by 15,

$\dfrac{75}{90} = \dfrac{75 ÷ 15}{90 ÷ 15} = \dfrac{5}{6}$.

Now, to get a fraction with denominator 42, we multiply both the numerator and denominator by 7,

$\dfrac{5}{6} = \dfrac{5 \times 7}{6 \times 7} = \dfrac{35}{42}$.

Hence, the equivalent fraction = $\dfrac{35}{42}$.

Write the missing numerals in the place-holders :

(i) $\dfrac{4}{9}$ = $\dfrac{\boxed{\phantom{63}}}{63}$

(ii) $\dfrac{3}{4}$ = $\dfrac{18}{\boxed{\phantom{63}}}$

(iii) $\dfrac{42}{70}$ = $\dfrac{3}{\boxed{\phantom{63}}}$

(iv) $\dfrac{52}{78}$ = $\dfrac{\boxed{\phantom{63}}}{6}$

Answer

(i) $\dfrac{4}{9}$ = $\dfrac{\boxed{\phantom{63}}}{63}$

$\dfrac{4}{9} = \dfrac{4 \times 7}{9 \times 7} = \dfrac{28}{63}$

Hence, $\dfrac{4}{9}$ = $\dfrac{\boxed{28}}{63}$.

(ii) $\dfrac{3}{4}$ = $\dfrac{18}{\boxed{\phantom{63}}}$

$\dfrac{3}{4} = \dfrac{3 \times 6}{4 \times 6} = \dfrac{18}{24}$

Hence, $\dfrac{3}{4}$ = $\dfrac{18}{\boxed{24}}$.

(iii) $\dfrac{42}{70}$ = $\dfrac{3}{\boxed{\phantom{63}}}$

Simplifying the fraction,

$\dfrac{42}{70} = \dfrac{42 ÷ 14}{70 ÷ 14} = \dfrac{3}{5}$

Hence, $\dfrac{42}{70}$ = $\dfrac{3}{\boxed{5}}$.

(iv) $\dfrac{52}{78}$ = $\dfrac{\boxed{\phantom{63}}}{6}$

Simplifying the fraction,

$\dfrac{52}{78} = \dfrac{52 ÷ 13}{78 ÷ 13} = \dfrac{4}{6}$

Hence, $\dfrac{52}{78}$ = $\dfrac{\boxed{4}}{6}$.

What fraction of an hour is 25 minutes?

Answer

Given minutes = 25 minutes

Total minutes = 60 minutes

Fraction = $\dfrac{\text{Given minutes}}{\text{Total minutes}} = \dfrac{25}{60}$.

Hence, required fraction = $\dfrac{25}{60}$.

What fraction of a day is 9 hours?

Answer

Given hours = 9 hours

Total hours = 24 hours

Fraction = $\dfrac{\text{Given hours}}{\text{Total hours}} = \dfrac{9}{24}$.

Hence, required fraction = $\dfrac{9}{24}$.