Fractions — Exercise 6(A)

For each expression, given below, write a fraction:

(i) 2 out of 7 = ...............

(ii) 5 out of 17 = ...............

(iii) three-fifths = ...............

Answer

As we know, 'out of' means division, so it is written as a fraction $\dfrac{\text{part}}{\text{whole}}$

(i) 2 out of 7

⇒ 2 out of 7 = $\dfrac{2}{7}$.

Hence, 2 out of 7 = $\dfrac{2}{7}$.

(ii) 5 out of 17

⇒ 5 out of 17 = $\dfrac{5}{17}$.

Hence, 5 out of 17 = $\dfrac{5}{17}$.

(iii) three-fifths

⇒ three-fifths = $\dfrac{3}{5}$.

Hence, three-fifths = $\dfrac{3}{5}$.

Fill in the blanks:

$\dfrac{5}{8}$ is ............... fraction.

Answer

In $\dfrac{5}{8}$, numerator (5) < denominator (8).

Hence, $\dfrac{5}{8}$ is a proper fraction.

Fill in the blanks:

$\dfrac{8}{5}$ is ............... fraction.

Answer

In $\dfrac{8}{5}$, numerator (8) > denominator (5).

Hence, $\dfrac{8}{5}$ is an improper fraction.

Fill in the blanks:

$\dfrac{15}{15}$ is ............... fraction.

Answer

In $\dfrac{15}{15}$, numerator (15) = denominator (15), i.e. numerator is not less than denominator.

Hence, $\dfrac{15}{15}$ is an improper fraction.

Fill in the blanks:

The value of $\dfrac{23}{23}$ = ............... .

Answer

$\dfrac{23}{23} = 23 \div 23 = 1$.

Hence, the value of $\dfrac{23}{23}$ is 1.

Fill in the blanks:

The value of $\dfrac{5}{5}$ = ...............

Answer

$\dfrac{5}{5} = 5 \div 5 = 1$.

Hence, the value of $\dfrac{5}{5}$ is 1.

Fill in the blanks:

$3\dfrac{3}{10}$ is ............... fraction.

Answer

$3\dfrac{3}{10}$ consists of a natural number 3 and a proper fraction $\dfrac{3}{10}$.

Hence, $3\dfrac{3}{10}$ is a mixed fraction.

Fill in the blanks:

$\dfrac{2}{15} \text{ and } \dfrac{7}{15}$ are ............... fractions.

Answer

$\dfrac{2}{15} \text{ and } \dfrac{7}{15}$ have the same denominator 15.

Hence, $\dfrac{2}{15} \text{ and } \dfrac{7}{15}$ are like fractions.

Fill in the blanks:

$\dfrac{23}{12} \text{ and } \dfrac{23}{15}$ are ............... fractions.

Answer

$\dfrac{23}{12} \text{ and } \dfrac{23}{15}$ have different denominators (12 and 15).

Hence, $\dfrac{23}{12} \text{ and } \dfrac{23}{15}$ are unlike fractions.

Fill in the blanks:

$\dfrac{6}{15} \text{ and } \dfrac{28}{70}$ are ............... fractions.

Answer

Reducing to lowest terms, $\dfrac{6}{15} = \dfrac{2}{5} \text{ and } \dfrac{28}{70} = \dfrac{2}{5}$.

Both are equal to $\dfrac{2}{5}$.

Hence, $\dfrac{6}{15} \text{ and } \dfrac{28}{70}$ are equal (equivalent) fractions.

Fill in the blanks:

$\dfrac{8}{24} \text{ and } \dfrac{8}{32}$ are not ............... fractions.

Answer

$\dfrac{8}{24} \text{ and } \dfrac{8}{32}$ have different denominators (24 and 32).

Hence, $\dfrac{8}{24} \text{ and } \dfrac{8}{32}$ are not like fractions.

Fill in the blanks:

$3\dfrac{2}{13} = \dfrac{3 \times 13 + ...............}{13}$ = ...............

Answer

$3\dfrac{2}{13}$

$\Rightarrow 3\dfrac{2}{13} = \dfrac{3 \times 13 + 2}{13} \\[1em] = \dfrac{39 + 2}{13} \\[1em] = \dfrac{41}{13}.$

Hence, $3\dfrac{2}{13} = \dfrac{3 \times 13 + 2}{13} = \dfrac{41}{13}$.

Fill in the blanks:

$4\dfrac{3}{5}$ = ............... = ...............

Answer

$4\dfrac{3}{5}$

$\Rightarrow 4\dfrac{3}{5} = \dfrac{4 \times 5 + 3}{5} \\[1em] = \dfrac{20 + 3}{5} \\[1em] = \dfrac{23}{5}.$

Hence, $4\dfrac{3}{5} = \dfrac{4 \times 5 + 3}{5} = \dfrac{23}{5}$.

From the following fractions, separate (i) proper fractions and (ii) improper fractions:

$\dfrac{2}{9}, \dfrac{4}{3}, \dfrac{7}{15}, \dfrac{11}{20}, \dfrac{20}{11}, \dfrac{18}{23}, \dfrac{27}{35}$.

Answer

A fraction is proper if numerator < denominator, and improper if numerator ≥ denominator.

(i) Proper fractions (numerator < denominator) are:

$\dfrac{2}{9}, \dfrac{7}{15}, \dfrac{11}{20}, \dfrac{18}{23} \text{ and } \dfrac{27}{35}$.

(ii) Improper fractions (numerator > denominator) are:

$\dfrac{4}{3} \text{ and } \dfrac{20}{11}$.

Hence, proper fractions are $\dfrac{2}{9}, \dfrac{7}{15}, \dfrac{11}{20}, \dfrac{18}{23}, \dfrac{27}{35}$ and improper fractions are $\dfrac{4}{3}, \dfrac{20}{11}$.

Change the following mixed fractions to improper fractions:

$2\dfrac{1}{5}$

Answer

$2\dfrac{1}{5}$

$\Rightarrow 2\dfrac{1}{5} = \dfrac{2 \times 5 + 1}{5} \\[1em] = \dfrac{10 + 1}{5} \\[1em] = \dfrac{11}{5}.$

Hence, $2\dfrac{1}{5} = \dfrac{11}{5}$.

Change the following mixed fractions to improper fractions:

$3\dfrac{1}{4}$

Answer

$3\dfrac{1}{4}$

$\Rightarrow 3\dfrac{1}{4} = \dfrac{3 \times 4 + 1}{4} \\[1em] = \dfrac{12 + 1}{4} \\[1em] = \dfrac{13}{4}.$

Hence, $3\dfrac{1}{4} = \dfrac{13}{4}$.

Change the following mixed fractions to improper fractions:

$7\dfrac{1}{8}$

Answer

$7\dfrac{1}{8}$

$\Rightarrow 7\dfrac{1}{8} = \dfrac{7 \times 8 + 1}{8} \\[1em] = \dfrac{56 + 1}{8} \\[1em] = \dfrac{57}{8}.$

Hence, $7\dfrac{1}{8} = \dfrac{57}{8}$.

Change the following mixed fractions to improper fractions:

$2\dfrac{1}{11}$

Answer

$2\dfrac{1}{11}$

$\Rightarrow 2\dfrac{1}{11} = \dfrac{2 \times 11 + 1}{11} \\[1em] = \dfrac{22 + 1}{11} \\[1em] = \dfrac{23}{11}.$

Hence, $2\dfrac{1}{11} = \dfrac{23}{11}$.

Change the following improper fractions to mixed fractions:

$\dfrac{100}{17}$

Answer

$\dfrac{100}{17}$

Dividing 100 by 17, we get quotient = 5 and remainder = 15.

$\Rightarrow \dfrac{100}{17} = 5\dfrac{15}{17}.$

Hence, $\dfrac{100}{17} = 5\dfrac{15}{17}$.

Change the following improper fractions to mixed fractions:

$\dfrac{81}{11}$

Answer

$\dfrac{81}{11}$

Dividing 81 by 11, we get quotient = 7 and remainder = 4.

$\Rightarrow \dfrac{81}{11} = 7\dfrac{4}{11}.$

Hence, $\dfrac{81}{11} = 7\dfrac{4}{11}$.

Change the following improper fractions to mixed fractions:

$\dfrac{209}{7}$

Answer

$\dfrac{209}{7}$

Dividing 209 by 7, we get quotient = 29 and remainder = 6.

$\Rightarrow \dfrac{209}{7} = 29\dfrac{6}{7}.$

Hence, $\dfrac{209}{7} = 29\dfrac{6}{7}$.

Change the following improper fractions to mixed fractions:

$\dfrac{113}{15}$

Answer

$\dfrac{113}{15}$

Dividing 113 by 15, we get quotient = 7 and remainder = 8.

$\Rightarrow \dfrac{113}{15} = 7\dfrac{8}{15}.$

Hence, $\dfrac{113}{15} = 7\dfrac{8}{15}$.

Change the following groups of fractions to like fractions:

$\dfrac{1}{3}, \dfrac{2}{5}, \dfrac{3}{4}, \dfrac{1}{6}$

Answer

$\dfrac{1}{3}, \dfrac{2}{5}, \dfrac{3}{4}, \dfrac{1}{6}$

LCM of 3, 5, 4 and 6 = 60.

$\Rightarrow \dfrac{1}{3} = \dfrac{1 \times 20}{3 \times 20} = \dfrac{20}{60} \\[1em] \Rightarrow \dfrac{2}{5} = \dfrac{2 \times 12}{5 \times 12} = \dfrac{24}{60} \\[1em] \Rightarrow \dfrac{3}{4} = \dfrac{3 \times 15}{4 \times 15} = \dfrac{45}{60} \\[1em] \Rightarrow \dfrac{1}{6} = \dfrac{1 \times 10}{6 \times 10} = \dfrac{10}{60}$

Hence, the required like fractions are $\dfrac{20}{60}, \dfrac{24}{60}, \dfrac{45}{60} \text{ and } \dfrac{10}{60}$.

Change the following groups of fractions to like fractions:

$\dfrac{5}{6}, \dfrac{7}{8}, \dfrac{11}{12}, \dfrac{3}{10}$

Answer

$\dfrac{5}{6}, \dfrac{7}{8}, \dfrac{11}{12}, \dfrac{3}{10}$

LCM of 6, 8, 12 and 10 = 120.

$\Rightarrow \dfrac{5}{6} = \dfrac{5 \times 20}{6 \times 20} = \dfrac{100}{120} \\[1em] \Rightarrow \dfrac{7}{8} = \dfrac{7 \times 15}{8 \times 15} = \dfrac{105}{120} \\[1em] \Rightarrow \dfrac{11}{12} = \dfrac{11 \times 10}{12 \times 10} = \dfrac{110}{120} \\[1em] \Rightarrow \dfrac{3}{10} = \dfrac{3 \times 12}{10 \times 12} = \dfrac{36}{120}$

Hence, the required like fractions are $\dfrac{100}{120}, \dfrac{105}{120}, \dfrac{110}{120} \text{ and } \dfrac{36}{120}$.

Change the following groups of fractions to like fractions:

$\dfrac{2}{7}, \dfrac{7}{8}, \dfrac{5}{14}, \dfrac{9}{16}$

Answer

$\dfrac{2}{7}, \dfrac{7}{8}, \dfrac{5}{14}, \dfrac{9}{16}$

LCM of 7, 8, 14 and 16 = 112.

$\Rightarrow \dfrac{2}{7} = \dfrac{2 \times 16}{7 \times 16} = \dfrac{32}{112} \\[1em] \Rightarrow \dfrac{7}{8} = \dfrac{7 \times 14}{8 \times 14} = \dfrac{98}{112} \\[1em] \Rightarrow \dfrac{5}{14} = \dfrac{5 \times 8}{14 \times 8} = \dfrac{40}{112} \\[1em] \Rightarrow \dfrac{9}{16} = \dfrac{9 \times 7}{16 \times 7} = \dfrac{63}{112}$

Hence, the required like fractions are $\dfrac{32}{112}, \dfrac{98}{112}, \dfrac{40}{112} \text{ and } \dfrac{63}{112}$.