Fractions — Exercise 3(A)

Classify each fraction, given below, as decimal or vulgar fraction, proper or improper fraction and mixed fraction:

(i) $\dfrac{3}{5}$

(ii) $\dfrac{11}{10}$

(iii) $\dfrac{13}{20}$

(iv) $\dfrac{18}{7}$

(v) $3\dfrac{2}{9}$

Answer

(i) $\dfrac{3}{5}$ : The denominator is neither 10 nor any higher power of 10. Thus it is a vulgar fraction. Since the denominator (5) is greater than the numerator (3), it is a proper fraction.

(ii) $\dfrac{11}{10}$ : The denominator is 10, so it is a decimal fraction. Since the denominator (10) is less than the numerator (11), it is an improper fraction.

(iii) $\dfrac{13}{20}$ : The denominator is neither 10 nor any higher power of 10. Thus it is a vulgar fraction. Since the denominator (20) is greater than the numerator (13), it is a proper fraction.

(iv) $\dfrac{18}{7}$ : The denominator is neither 10 nor any higher power of 10, so it is a vulgar fraction. Since the denominator (7) is less than the numerator (18), it is an improper fraction.

(v) $3\dfrac{2}{9}$ : The denominator is neither 10 nor any higher power of 10, so it is a vulgar fraction. It consists of a natural number and a proper fraction, so it is a mixed fraction.

Express the following improper fractions as mixed fractions:

(i) $\dfrac{18}{5}$

(ii) $\dfrac{7}{4}$

(iii) $\dfrac{25}{6}$

(iv) $\dfrac{38}{5}$

(v) $\dfrac{22}{5}$

Answer

(i) Dividing 18 by 5 :

$\begin{array}{l} 5\overline{\smash{\big)} \phantom{)}18\smash{\big(}} 3 \\ \phantom{a}\underline{-15} \\ \phantom{+()}3 \\ \end{array}$

Quotient = 3 and remainder = 3.

$\dfrac{18}{5} = 3\dfrac{3}{5}$

Hence, $\dfrac{18}{5} = 3\dfrac{3}{5}$

(ii) Dividing 7 by 4 :

$\begin{array}{l} 4\overline{\smash{\big)} \phantom{)}7\smash{\big(}} 1 \\ \phantom{a}\underline{-4} \\ \phantom{+)}3 \\ \end{array}$

Quotient = 1 and remainder = 3.

$\dfrac{7}{4} = 1\dfrac{3}{4}$

Hence, $\dfrac{7}{4} = 1\dfrac{3}{4}$

(iii) Dividing 25 by 6 :

$\begin{array}{l} 6\overline{\smash{\big)} \phantom{)}25\smash{\big(}} 4 \\ \phantom{a}\underline{-24} \\ \phantom{+))}1 \\ \end{array}$

Quotient = 4 and remainder = 1.

$\dfrac{25}{6} = 4\dfrac{1}{6}$

Hence, $\dfrac{25}{6} = 4\dfrac{1}{6}$

(iv) Dividing 38 by 5 :

$\begin{array}{l} 5\overline{\smash{\big)} \phantom{)}38\smash{\big(}} 7 \\ \phantom{a}\underline{-35} \\ \phantom{+((}3 \\ \end{array}$

Quotient = 7 and remainder = 3.

$\dfrac{38}{5} = 7\dfrac{3}{5}$

Hence, $\dfrac{38}{5} = 7\dfrac{3}{5}$

(v) Dividing 22 by 5 :

$\begin{array}{l} 5\overline{\smash{\big)} \phantom{)}22\smash{\big(}} 4 \\ \phantom{a}\underline{-20} \\ \phantom{+((}2 \\ \end{array}$

Quotient = 4 and remainder = 2.

$\dfrac{22}{5} = 4\dfrac{2}{5}$

Hence, $\dfrac{22}{5} = 4\dfrac{2}{5}$

Express the following mixed fractions as improper fractions:

(i) $2\dfrac{4}{9}$

(ii) $7\dfrac{5}{13}$

(iii) $3\dfrac{1}{4}$

(iv) $2\dfrac{5}{48}$

(v) $12\dfrac{7}{11}$

Answer

(i) $2\dfrac{4}{9} = \dfrac{2 \times 9 + 4}{9} = \dfrac{18 + 4}{9} = \dfrac{22}{9}$

Hence, $2\dfrac{4}{9} = \dfrac{22}{9}$

(ii) $7\dfrac{5}{13} = \dfrac{7 \times 13 + 5}{13} = \dfrac{91 + 5}{13} = \dfrac{96}{13}$

Hence, $7\dfrac{5}{13} = \dfrac{96}{13}$

(iii) $3\dfrac{1}{4} = \dfrac{3 \times 4 + 1}{4} = \dfrac{12 + 1}{4} = \dfrac{13}{4}$

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

(iv) $2\dfrac{5}{48} = \dfrac{2 \times 48 + 5}{48} = \dfrac{96 + 5}{48} = \dfrac{101}{48}$

Hence, $2\dfrac{5}{48} = \dfrac{101}{48}$

(v) $12\dfrac{7}{11} = \dfrac{12 \times 11 + 7}{11} = \dfrac{132 + 7}{11} = \dfrac{139}{11}$

Hence, $12\dfrac{7}{11} = \dfrac{139}{11}$

Reduce the given fractions to lowest terms:

(i) $\dfrac{8}{18}$

(ii) $\dfrac{27}{36}$

(iii) $\dfrac{18}{42}$

(iv) $\dfrac{35}{75}$

(v) $\dfrac{18}{45}$

Answer

(i) HCF of 8 and 18 = 2.

$\dfrac{8}{18} = \dfrac{8 \div 2}{18 \div 2} = \dfrac{4}{9}$

Hence, $\dfrac{8}{18} = \dfrac{4}{9}$

(ii) HCF of 27 and 36 = 9.

$\dfrac{27}{36} = \dfrac{27 \div 9}{36 \div 9} = \dfrac{3}{4}$

Hence, $\dfrac{27}{36} = \dfrac{3}{4}$

(iii) HCF of 18 and 42 = 6.

$\dfrac{18}{42} = \dfrac{18 \div 6}{42 \div 6} = \dfrac{3}{7}$

Hence, $\dfrac{18}{42} = \dfrac{3}{7}$

(iv) HCF of 35 and 75 = 5.

$\dfrac{35}{75} = \dfrac{35 \div 5}{75 \div 5} = \dfrac{7}{15}$

Hence, $\dfrac{35}{75} = \dfrac{7}{15}$

(v) HCF of 18 and 45 = 9.

$\dfrac{18}{45} = \dfrac{18 \div 9}{45 \div 9} = \dfrac{2}{5}$

Hence, $\dfrac{18}{45} = \dfrac{2}{5}$

State true or false:

(i) $\dfrac{30}{40} \text{ and } \dfrac{12}{16}$ are equivalent fractions.

(ii) $\dfrac{10}{25} \text{ and } \dfrac{25}{10}$ are equivalent fractions.

(iii) $\dfrac{35}{49}, \dfrac{20}{28}, \dfrac{45}{63} \text{ and } \dfrac{100}{140}$ are equivalent fractions.

Answer

(i) $\dfrac{30}{40} = \dfrac{30 \div 10}{40 \div 10} = \dfrac{3}{4} \text{ and } \dfrac{12}{16} = \dfrac{12 \div 4}{16 \div 4} = \dfrac{3}{4}$

Both reduce to $\dfrac{3}{4}$, so they are equivalent fractions.

Hence, the statement is True.

(ii) $\dfrac{10}{25} = \dfrac{10 \div 5}{25 \div 5} = \dfrac{2}{5} \text{ and } \dfrac{25}{10} = \dfrac{25 \div 5}{10 \div 5} = \dfrac{5}{2}$

Since $\dfrac{2}{5} ≠ \dfrac{5}{2}$, they are not equivalent fractions.

Hence, the statement is False.

(iii) $\dfrac{35}{49} = \dfrac{5}{7}, \dfrac{20}{28} = \dfrac{5}{7}, \dfrac{45}{63} = \dfrac{5}{7} \text{ and } \dfrac{100}{140} = \dfrac{5}{7}$

All four reduce to $\dfrac{5}{7}$, so they are equivalent fractions.

Hence, the statement is True.

Distinguish each of the fractions, given below, as a simple fraction or a complex fraction:

(i) $\dfrac{0}{8}$

(ii) $\dfrac{3}{8}$

(iii) $\dfrac{5}{7}$

(iv) $\dfrac{3\dfrac{3}{5}}{18}$

(v) $\dfrac{6}{2\dfrac{2}{5}}$

(vi) $\dfrac{3\dfrac{1}{3}}{7\dfrac{2}{7}}$

(vii) $\dfrac{5\dfrac{2}{9}}{5}$

(viii) $\dfrac{8}{0}$

Answer

A simple fraction has both numerator and denominator as whole numbers (with denominator ≠ 0); a complex fraction has a fraction in its numerator or denominator or both.

(i) $\dfrac{0}{8}$ : both terms are whole numbers and denominator ≠ 0, so it is a Simple fraction.

(ii) $\dfrac{3}{8}$ : both terms are whole numbers, so it is a Simple fraction.

(iii) $\dfrac{5}{7}$ : both terms are whole numbers, so it is a Simple fraction.

(iv) $\dfrac{3\dfrac{3}{5}}{18}$ : the numerator is a fraction, so it is a Complex fraction.

(v) $\dfrac{6}{2\dfrac{2}{5}}$ : the denominator is a fraction, so it is a Complex fraction.

(vi) $\dfrac{3\dfrac{1}{3}}{7\dfrac{2}{7}}$ : both numerator and denominator are fractions, so it is a Complex fraction.

(vii) $\dfrac{5\dfrac{2}{9}}{5}$ : the numerator is a fraction, so it is a Complex fraction.

(viii) $\dfrac{8}{0}$ : division by 0 is not defined, so it is Neither a simple nor a complex fraction.