Q1. Evaluate
(i) $2.2 + 1\dfrac{4}{7} - 4.5 + 2\dfrac{8}{35}$
(ii) $0.8 \times \dfrac{7}{12} \div \dfrac{5}{24}$
(iii) $4.375 \div 2\dfrac{1}{11}$ of $2\dfrac{9}{23}$
(iv) $11 \div 1\dfrac{9}{22} \times 5\dfrac{7}{11}$
(v) $(5 \div 2.25) \div (9 \div 2.25)$
(vi) $3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \overline{\dfrac{1}{4} - \dfrac{1}{6}})\rbrace$
(vii) $3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \overline{\dfrac{1}{4} - \dfrac{1}{6}})]$
Show Answer
(i) $ 2.2 + 1\dfrac{4}{7} - 4.5 + 2\dfrac{8}{35}$
$\Rightarrow \dfrac{22}{10} + \dfrac{11}{7} - \dfrac{45}{10} + \dfrac{78}{35}$
$\Rightarrow$ L.C.M. of $10, 7, 35, = 70$
$\Rightarrow \dfrac{154 + 110 - 315 + 156}{70}$
$\Rightarrow \dfrac{264 + 156 - 315}{70}$
$\Rightarrow \dfrac{420 - 315}{70}$
$\Rightarrow \dfrac{105}{70} \Rightarrow \dfrac{21}{14} \Rightarrow \dfrac{3}{2} \Rightarrow 1\dfrac{1}{2}$ Ans
(ii) $0.8 \times \dfrac{7}{12} \div \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{12} \div \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{12} \times \dfrac{24}{5}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7 \times 2}{1 \times 5}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{14}{5}$
$\Rightarrow \dfrac{112}{50} \Rightarrow \dfrac{56}{25} \Rightarrow 2\dfrac{6}{25}$ Ans
(iii) $ 4.375 \div 2\dfrac{1}{11}$ of $2\dfrac{9}{23}$
$\Rightarrow \dfrac{4375}{1000} \div \dfrac{23}{11}$ of $\dfrac{55}{23}$
$\Rightarrow \dfrac{4375}{1000} \div \dfrac{23}{11} \times \dfrac{55}{23}$
$\Rightarrow \dfrac{4375}{1000} \div 5$
$\Rightarrow \dfrac{4375}{1000} \times \dfrac{1}{5}$
$\Rightarrow \dfrac{875}{1000} \Rightarrow \dfrac{175}{200} \Rightarrow \dfrac{35}{40} \Rightarrow \dfrac{7}{8}$ Ans
(iv) $ 11 \div 1\dfrac{9}{22} \times 5\dfrac{7}{11}$
$\Rightarrow 11 \div \dfrac{31}{22} \times \dfrac{62}{11}$
$\Rightarrow 11 \times \dfrac{22}{31} \times \dfrac{62}{11}$
$\Rightarrow 11 \times 2 \times 2 \Rightarrow 44$ Ans
(v) $(5 \div 2.25) \div (9 \div 2.25)$
$\Rightarrow (5 \div \dfrac{225}{100}) \div (9 \div \dfrac{225}{100})$
$\Rightarrow (5 \times \dfrac{100}{225}) \div (9 \times \dfrac{100}{225})$
$\Rightarrow (\dfrac{100}{45}) \div (\dfrac{100}{25})$
$\Rightarrow \dfrac{20}{9} \div 4$
$\Rightarrow \dfrac{20}{9} \times \dfrac{1}{4}$
$\Rightarrow \dfrac{5}{9}$ Ans
(vi) $3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \overline{\dfrac{1}{4} - \dfrac{1}{6}})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{3 - 2}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{1}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{4 - 1}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{3}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{4})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} \times \dfrac{1}{4}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{1}{3} \times \dfrac{1}{2}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{1}{6}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6} \times \dfrac{1}{6}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{36}\rbrace$
$\Rightarrow \dfrac{375}{100} \times \dfrac{36}{5}$
$\Rightarrow \dfrac{75}{25} \times 9$
$\Rightarrow 3 \times 9 \Rightarrow 27$ Ans
(vii) $3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{3 - 2}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \overline{\dfrac{1}{4} - \dfrac{1}{6}})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{3 - 2}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{1}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} \times (\dfrac{4 - 1}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} \times (\dfrac{3}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} \times \dfrac{1}{4}]$
$\Rightarrow 37.5 - [3.75 \div \dfrac{5}{6} \times \dfrac{1}{6}]$
$\Rightarrow 37.5 - [3.75 \div \dfrac{5}{36}]$
$\Rightarrow 37.5 - [\dfrac{375}{100} \times \dfrac{36}{5}]$
$\Rightarrow 37.5 - [\dfrac{375}{100} \times \dfrac{36}{5}]$
$\Rightarrow 37.5 - [\dfrac{75}{25} \times 9]$
$\Rightarrow 37.5 - [3 \times 9]$
$\Rightarrow 37.5 - 27$
$\Rightarrow 37.5 - 27 \Rightarrow 10.5$ Ans
$\Rightarrow \dfrac{22}{10} + \dfrac{11}{7} - \dfrac{45}{10} + \dfrac{78}{35}$
$\Rightarrow$ L.C.M. of $10, 7, 35, = 70$
$\Rightarrow \dfrac{154 + 110 - 315 + 156}{70}$
$\Rightarrow \dfrac{264 + 156 - 315}{70}$
$\Rightarrow \dfrac{420 - 315}{70}$
$\Rightarrow \dfrac{105}{70} \Rightarrow \dfrac{21}{14} \Rightarrow \dfrac{3}{2} \Rightarrow 1\dfrac{1}{2}$ Ans
(ii) $0.8 \times \dfrac{7}{12} \div \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{12} \div \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{12} \times \dfrac{24}{5}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7 \times 2}{1 \times 5}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{14}{5}$
$\Rightarrow \dfrac{112}{50} \Rightarrow \dfrac{56}{25} \Rightarrow 2\dfrac{6}{25}$ Ans
(iii) $ 4.375 \div 2\dfrac{1}{11}$ of $2\dfrac{9}{23}$
$\Rightarrow \dfrac{4375}{1000} \div \dfrac{23}{11}$ of $\dfrac{55}{23}$
$\Rightarrow \dfrac{4375}{1000} \div \dfrac{23}{11} \times \dfrac{55}{23}$
$\Rightarrow \dfrac{4375}{1000} \div 5$
$\Rightarrow \dfrac{4375}{1000} \times \dfrac{1}{5}$
$\Rightarrow \dfrac{875}{1000} \Rightarrow \dfrac{175}{200} \Rightarrow \dfrac{35}{40} \Rightarrow \dfrac{7}{8}$ Ans
(iv) $ 11 \div 1\dfrac{9}{22} \times 5\dfrac{7}{11}$
$\Rightarrow 11 \div \dfrac{31}{22} \times \dfrac{62}{11}$
$\Rightarrow 11 \times \dfrac{22}{31} \times \dfrac{62}{11}$
$\Rightarrow 11 \times 2 \times 2 \Rightarrow 44$ Ans
(v) $(5 \div 2.25) \div (9 \div 2.25)$
$\Rightarrow (5 \div \dfrac{225}{100}) \div (9 \div \dfrac{225}{100})$
$\Rightarrow (5 \times \dfrac{100}{225}) \div (9 \times \dfrac{100}{225})$
$\Rightarrow (\dfrac{100}{45}) \div (\dfrac{100}{25})$
$\Rightarrow \dfrac{20}{9} \div 4$
$\Rightarrow \dfrac{20}{9} \times \dfrac{1}{4}$
$\Rightarrow \dfrac{5}{9}$ Ans
(vi) $3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \overline{\dfrac{1}{4} - \dfrac{1}{6}})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{3 - 2}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{1}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{4 - 1}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{3}{12})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{4})\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{2}{3} \times \dfrac{1}{4}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{1}{3} \times \dfrac{1}{2}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6}$ of $\dfrac{1}{6}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{6} \times \dfrac{1}{6}\rbrace$
$\Rightarrow 3.75 \div \lbrace\dfrac{5}{36}\rbrace$
$\Rightarrow \dfrac{375}{100} \times \dfrac{36}{5}$
$\Rightarrow \dfrac{75}{25} \times 9$
$\Rightarrow 3 \times 9 \Rightarrow 27$ Ans
(vii) $3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{3 - 2}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \overline{\dfrac{1}{4} - \dfrac{1}{6}})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{3 - 2}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} (\dfrac{1}{3} - \dfrac{1}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} \times (\dfrac{4 - 1}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} \times (\dfrac{3}{12})]$
$\Rightarrow 3.75 - [3.75 \div \dfrac{5}{6}$ of $\dfrac{2}{3} \times \dfrac{1}{4}]$
$\Rightarrow 37.5 - [3.75 \div \dfrac{5}{6} \times \dfrac{1}{6}]$
$\Rightarrow 37.5 - [3.75 \div \dfrac{5}{36}]$
$\Rightarrow 37.5 - [\dfrac{375}{100} \times \dfrac{36}{5}]$
$\Rightarrow 37.5 - [\dfrac{375}{100} \times \dfrac{36}{5}]$
$\Rightarrow 37.5 - [\dfrac{75}{25} \times 9]$
$\Rightarrow 37.5 - [3 \times 9]$
$\Rightarrow 37.5 - 27$
$\Rightarrow 37.5 - 27 \Rightarrow 10.5$ Ans
Q2. Evaluate
(i) $0.8 \div \dfrac{7}{12} \times \dfrac{5}{24}$
(ii) $0.5 + 0.5 \div (0.5 + 0.5 \div 1)$
(iii) $1 \div \lbrace1 + 1 \div (1 + 1 \div 2)\rbrace$
(iv) $0.4 \times \dfrac{7}{3} \div \dfrac{15}{8}$ of $(\dfrac{7}{5} - \dfrac{4}{3})$
(v) $3\dfrac{1}{3} \div 2.5 \times 0.75 \div \dfrac{1}{3}$ of $20 \times \dfrac{7}{6}$
(vi) $1 + \dfrac{1}{1 + \dfrac{1}{2}}$
(vii) $1 - \dfrac{1}{1 - \dfrac{1}{1 - \dfrac{1}{3}}}$
Show Answer
(i) $0.8 \div \dfrac{7}{12} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{12}{7} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{5} \times \dfrac{6}{7} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{48}{35} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{2}{7}$ Ans
(ii) $0.5 + 0.5 \div (0.5 + 0.5 \div 1)$
$\Rightarrow 0.5 + 0.5 \div (0.5 + 0.5 \times \dfrac{1}{1})$
$\Rightarrow 0.5 + 0.5 \div (0.5 + 0.5)$
$\Rightarrow 0.5 + 0.5 \div 1$
$\Rightarrow 0.5 + 0.5 \times \dfrac{1}{1}$
$\Rightarrow 0.5 + 0.5 = 1$ Ans
(iii) $1 \div \lbrace1 + 1 \div (1 + 1 \div 2)\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div (1 + 1 \times \dfrac{1}{2})\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div (1 + \dfrac{1}{2})\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div (\dfrac{2 + 1}{2})\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div \dfrac{3}{2}\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \times \dfrac{2}{3}\rbrace$
$\Rightarrow 1 \div \lbrace1 + \dfrac{2}{3}\rbrace$
$\Rightarrow 1 \div \lbrace\dfrac{3 + 2}{3}\rbrace$
$\Rightarrow 1 \div \lbrace\dfrac{5}{3}\rbrace$
$\Rightarrow 1 \times \dfrac{3}{5} = \dfrac{3}{5}$ Ans
(iv) $0.4 \times \dfrac{7}{3} \div \dfrac{15}{8}$ of $(\dfrac{7}{5} - \dfrac{4}{3})$
$\Rightarrow 0.4 \times \dfrac{7}{3} \div \dfrac{15}{8}$ of $(\dfrac{21 - 20}{15})$
$\Rightarrow 0.4 \times \dfrac{7}{3} \div \dfrac{15}{8} \times \dfrac{1}{15}$
$\Rightarrow 0.4 \times \dfrac{7}{3} \div \dfrac{1}{8}$
$\Rightarrow 0.4 \times \dfrac{7}{3} \times \dfrac{8}{1}$
$\Rightarrow \dfrac{4}{10} \times \dfrac{56}{3}$
$\Rightarrow \dfrac{4}{5} \times \dfrac{28}{3}$
$\Rightarrow \dfrac{112}{15} = 7\dfrac{7}{15}$ Ans
(v) $3\dfrac{1}{3} \div 2.5 \times 0.75 \div \dfrac{1}{3}$ of $20 \times \dfrac{7}{6}$
$\Rightarrow \dfrac{10}{3} \div \dfrac{25}{10} \times \dfrac{75}{100} \div \dfrac{1}{3} \times 20 \times \dfrac{7}{6}$
$\Rightarrow \dfrac{10}{3} \div \dfrac{25}{10} \times \dfrac{75}{100} \div \dfrac{20}{3} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{10}{3} \times \dfrac{10}{25} \times \dfrac{75}{100} \times \dfrac{3}{20} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{100}{75} \times \dfrac{225}{2000} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{3}{20} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{7}{20 \times 2} = \dfrac{7}{40}$ Ans
(vi) $1 + \dfrac{1}{1 + \dfrac{1}{2}}$
$\Rightarrow 1 + \dfrac{1}{\dfrac{2 + 1}{2}}$
$\Rightarrow 1 + \dfrac{1}{\dfrac{3}{2}}$
$\Rightarrow 1 + \dfrac{2}{3}$
$\Rightarrow \dfrac{3 + 2}{3}$
$\Rightarrow \dfrac{5}{3} \Rightarrow 1\dfrac{2}{3}$ Ans
(vii) $1 - \dfrac{1}{1 - \dfrac{1}{1 - \dfrac{1}{3}}}$
$\Rightarrow 1 - \dfrac{1}{1 - \dfrac{1}{\dfrac{3 - 1}{3}}}$
$\Rightarrow 1 - \dfrac{1}{1 - \dfrac{1}{\dfrac{2}{3}}}$
$\Rightarrow 1 - \dfrac{1}{1 - \dfrac{3}{2}}$
$\Rightarrow 1 - \dfrac{1}{\dfrac{2 - 3}{2}}$
$\Rightarrow 1 - \dfrac{1}{-\dfrac{1}{2}}$
$\Rightarrow 1 + \dfrac{2}{1}$
$\Rightarrow \dfrac{1 + 2}{1} \Rightarrow \dfrac{3}{1} = 3$ Ans
$\Rightarrow \dfrac{8}{10} \times \dfrac{12}{7} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{8}{5} \times \dfrac{6}{7} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{48}{35} \times \dfrac{5}{24}$
$\Rightarrow \dfrac{2}{7}$ Ans
(ii) $0.5 + 0.5 \div (0.5 + 0.5 \div 1)$
$\Rightarrow 0.5 + 0.5 \div (0.5 + 0.5 \times \dfrac{1}{1})$
$\Rightarrow 0.5 + 0.5 \div (0.5 + 0.5)$
$\Rightarrow 0.5 + 0.5 \div 1$
$\Rightarrow 0.5 + 0.5 \times \dfrac{1}{1}$
$\Rightarrow 0.5 + 0.5 = 1$ Ans
(iii) $1 \div \lbrace1 + 1 \div (1 + 1 \div 2)\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div (1 + 1 \times \dfrac{1}{2})\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div (1 + \dfrac{1}{2})\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div (\dfrac{2 + 1}{2})\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \div \dfrac{3}{2}\rbrace$
$\Rightarrow 1 \div \lbrace1 + 1 \times \dfrac{2}{3}\rbrace$
$\Rightarrow 1 \div \lbrace1 + \dfrac{2}{3}\rbrace$
$\Rightarrow 1 \div \lbrace\dfrac{3 + 2}{3}\rbrace$
$\Rightarrow 1 \div \lbrace\dfrac{5}{3}\rbrace$
$\Rightarrow 1 \times \dfrac{3}{5} = \dfrac{3}{5}$ Ans
(iv) $0.4 \times \dfrac{7}{3} \div \dfrac{15}{8}$ of $(\dfrac{7}{5} - \dfrac{4}{3})$
$\Rightarrow 0.4 \times \dfrac{7}{3} \div \dfrac{15}{8}$ of $(\dfrac{21 - 20}{15})$
$\Rightarrow 0.4 \times \dfrac{7}{3} \div \dfrac{15}{8} \times \dfrac{1}{15}$
$\Rightarrow 0.4 \times \dfrac{7}{3} \div \dfrac{1}{8}$
$\Rightarrow 0.4 \times \dfrac{7}{3} \times \dfrac{8}{1}$
$\Rightarrow \dfrac{4}{10} \times \dfrac{56}{3}$
$\Rightarrow \dfrac{4}{5} \times \dfrac{28}{3}$
$\Rightarrow \dfrac{112}{15} = 7\dfrac{7}{15}$ Ans
(v) $3\dfrac{1}{3} \div 2.5 \times 0.75 \div \dfrac{1}{3}$ of $20 \times \dfrac{7}{6}$
$\Rightarrow \dfrac{10}{3} \div \dfrac{25}{10} \times \dfrac{75}{100} \div \dfrac{1}{3} \times 20 \times \dfrac{7}{6}$
$\Rightarrow \dfrac{10}{3} \div \dfrac{25}{10} \times \dfrac{75}{100} \div \dfrac{20}{3} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{10}{3} \times \dfrac{10}{25} \times \dfrac{75}{100} \times \dfrac{3}{20} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{100}{75} \times \dfrac{225}{2000} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{3}{20} \times \dfrac{7}{6}$
$\Rightarrow \dfrac{7}{20 \times 2} = \dfrac{7}{40}$ Ans
(vi) $1 + \dfrac{1}{1 + \dfrac{1}{2}}$
$\Rightarrow 1 + \dfrac{1}{\dfrac{2 + 1}{2}}$
$\Rightarrow 1 + \dfrac{1}{\dfrac{3}{2}}$
$\Rightarrow 1 + \dfrac{2}{3}$
$\Rightarrow \dfrac{3 + 2}{3}$
$\Rightarrow \dfrac{5}{3} \Rightarrow 1\dfrac{2}{3}$ Ans
(vii) $1 - \dfrac{1}{1 - \dfrac{1}{1 - \dfrac{1}{3}}}$
$\Rightarrow 1 - \dfrac{1}{1 - \dfrac{1}{\dfrac{3 - 1}{3}}}$
$\Rightarrow 1 - \dfrac{1}{1 - \dfrac{1}{\dfrac{2}{3}}}$
$\Rightarrow 1 - \dfrac{1}{1 - \dfrac{3}{2}}$
$\Rightarrow 1 - \dfrac{1}{\dfrac{2 - 3}{2}}$
$\Rightarrow 1 - \dfrac{1}{-\dfrac{1}{2}}$
$\Rightarrow 1 + \dfrac{2}{1}$
$\Rightarrow \dfrac{1 + 2}{1} \Rightarrow \dfrac{3}{1} = 3$ Ans
Q3. Evaluate:
(i) $0.8 \times 0.7 - 0.5 \times 0.3 + 0.16 \div 0.04$
(ii) $4.4 - [2 - 0.64 \lbrace3 - 1.2 - (0.63 - 0.03)\rbrace]$
(iii) $\dfrac{9.75 \text{ of } 3.6}{0.039} + \dfrac{3.75}{0.015}$
(iv) $\dfrac{0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002}{0.12 \times 0.12}$
Show Answer
(i) $0.8 \times 0.7 - 0.5 \times 0.3 + 0.16 \div 0.04$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{10} - \dfrac{5}{10} \times \dfrac{3}{10} + \dfrac{16}{100} \div \dfrac{4}{100}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{10} - \dfrac{5}{10} \times \dfrac{3}{10} + \dfrac{16}{100} \times \dfrac{100}{4}$
$\Rightarrow \dfrac{56}{100} - \dfrac{15}{100} + 4$
$\Rightarrow (\dfrac{56}{100} + 4) - \dfrac{15}{100}$
$\Rightarrow \dfrac{56 + 400}{100} - \dfrac{15}{100}$
$\Rightarrow \dfrac{456}{100} - \dfrac{15}{100}$
$\Rightarrow \dfrac{456 - 15}{100}$
$\Rightarrow \dfrac{441}{100} = 4.41$ Ans
(ii) $4.4 - [2 - 0.64 \lbrace3 - 1.2 \div (0.63 - 0.03)\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \div (\dfrac{63}{100} - \dfrac{3}{100})\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \div (\dfrac{63 - 3}{100})\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \div \dfrac{60}{100}\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \times \dfrac{100}{60}\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3- \dfrac{10}{5}\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - 2\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \times 1]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100}]$
$\Rightarrow \dfrac{44}{10} - [\dfrac{200 - 64}{100}]$
$\Rightarrow \dfrac{44}{10} - [\dfrac{136}{100}]$
$\Rightarrow \dfrac{44}{10} - \dfrac{136}{100}$
$\Rightarrow \dfrac{440 - 136}{100}$
$\Rightarrow \dfrac{304}{100} \Rightarrow 3.04$ Ans
(iii) $\dfrac{9.75 \text{ of } 3.6}{0.039} + \dfrac{3.75}{0.015}$
$\Rightarrow \dfrac{\dfrac{975}{100} \text{ of } \dfrac{36}{10}}{\dfrac{39}{1000}} + \dfrac{\dfrac{375}{100}}{\dfrac{15}{1000}}$
$\Rightarrow \dfrac{975 \times 36 \times 1000}{39 \times 100 \times 10} + \dfrac{375 \times 1000}{15 \times 100}$
$\Rightarrow \dfrac{975 \times 36 \times 1000}{39 \times 100 \times 10} + \dfrac{375 \times 1000}{15 \times 100}$
$\Rightarrow \dfrac{975 \times 12}{13} + \dfrac{3750}{15}$
$\Rightarrow 75 \times 12 + 250$
$\Rightarrow 900 + 250 \Rightarrow 1150$ Ans
(iv) $\dfrac{0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002}{0.12 \times 0.12}$
$\Rightarrow \dfrac{0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002}{0.12 \times 0.12}$
First, solving the numerator part:
$\Rightarrow 0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002$
$\Rightarrow \dfrac{12}{100} \text{ of } (\dfrac{104}{10000} - \dfrac{2}{1000}) + \dfrac{36}{100} \times \dfrac{2}{1000}$
$\Rightarrow \dfrac{12}{100} \text{ of } (\dfrac{104 - 20}{10000}) + \dfrac{36}{100} \times \dfrac{2}{1000}$
$\Rightarrow \dfrac{12}{100} \times \dfrac{84}{10000} + \dfrac{36}{100} \times \dfrac{2}{1000}$
$\Rightarrow \dfrac{1008}{1000000} + \dfrac{72}{100000}$
$\Rightarrow \dfrac{1008 + 720}{1000000} \Rightarrow \dfrac{1728}{1000000}$
Now, solving the denominator part:
$\Rightarrow 0.12 \times 0.12$
$\Rightarrow \dfrac{12}{100} \times \dfrac{12}{100}$
$\Rightarrow \dfrac{12}{100} \times \dfrac{12}{100} \Rightarrow \dfrac{144}{10000}$
So, we have:
$\Rightarrow \dfrac{\dfrac{1728}{1000000}}{\dfrac{144}{10000}}$
$\Rightarrow \dfrac{1728 \times 10000}{144 \times 1000000}$
$\Rightarrow \dfrac{12}{100} \Rightarrow 0.12$ Ans
OR
$\dfrac{12}{100} = \dfrac{6}{50} = \dfrac{3}{25} = 0.12$ Ans
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{10} - \dfrac{5}{10} \times \dfrac{3}{10} + \dfrac{16}{100} \div \dfrac{4}{100}$
$\Rightarrow \dfrac{8}{10} \times \dfrac{7}{10} - \dfrac{5}{10} \times \dfrac{3}{10} + \dfrac{16}{100} \times \dfrac{100}{4}$
$\Rightarrow \dfrac{56}{100} - \dfrac{15}{100} + 4$
$\Rightarrow (\dfrac{56}{100} + 4) - \dfrac{15}{100}$
$\Rightarrow \dfrac{56 + 400}{100} - \dfrac{15}{100}$
$\Rightarrow \dfrac{456}{100} - \dfrac{15}{100}$
$\Rightarrow \dfrac{456 - 15}{100}$
$\Rightarrow \dfrac{441}{100} = 4.41$ Ans
(ii) $4.4 - [2 - 0.64 \lbrace3 - 1.2 \div (0.63 - 0.03)\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \div (\dfrac{63}{100} - \dfrac{3}{100})\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \div (\dfrac{63 - 3}{100})\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \div \dfrac{60}{100}\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - \dfrac{12}{10} \times \dfrac{100}{60}\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3- \dfrac{10}{5}\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \lbrace3 - 2\rbrace]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100} \times 1]$
$\Rightarrow \dfrac{44}{10} - [2 - \dfrac{64}{100}]$
$\Rightarrow \dfrac{44}{10} - [\dfrac{200 - 64}{100}]$
$\Rightarrow \dfrac{44}{10} - [\dfrac{136}{100}]$
$\Rightarrow \dfrac{44}{10} - \dfrac{136}{100}$
$\Rightarrow \dfrac{440 - 136}{100}$
$\Rightarrow \dfrac{304}{100} \Rightarrow 3.04$ Ans
(iii) $\dfrac{9.75 \text{ of } 3.6}{0.039} + \dfrac{3.75}{0.015}$
$\Rightarrow \dfrac{\dfrac{975}{100} \text{ of } \dfrac{36}{10}}{\dfrac{39}{1000}} + \dfrac{\dfrac{375}{100}}{\dfrac{15}{1000}}$
$\Rightarrow \dfrac{975 \times 36 \times 1000}{39 \times 100 \times 10} + \dfrac{375 \times 1000}{15 \times 100}$
$\Rightarrow \dfrac{975 \times 36 \times 1000}{39 \times 100 \times 10} + \dfrac{375 \times 1000}{15 \times 100}$
$\Rightarrow \dfrac{975 \times 12}{13} + \dfrac{3750}{15}$
$\Rightarrow 75 \times 12 + 250$
$\Rightarrow 900 + 250 \Rightarrow 1150$ Ans
(iv) $\dfrac{0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002}{0.12 \times 0.12}$
$\Rightarrow \dfrac{0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002}{0.12 \times 0.12}$
First, solving the numerator part:
$\Rightarrow 0.12 \text{ of } (0.0104 - 0.002) + 0.36 \times 0.002$
$\Rightarrow \dfrac{12}{100} \text{ of } (\dfrac{104}{10000} - \dfrac{2}{1000}) + \dfrac{36}{100} \times \dfrac{2}{1000}$
$\Rightarrow \dfrac{12}{100} \text{ of } (\dfrac{104 - 20}{10000}) + \dfrac{36}{100} \times \dfrac{2}{1000}$
$\Rightarrow \dfrac{12}{100} \times \dfrac{84}{10000} + \dfrac{36}{100} \times \dfrac{2}{1000}$
$\Rightarrow \dfrac{1008}{1000000} + \dfrac{72}{100000}$
$\Rightarrow \dfrac{1008 + 720}{1000000} \Rightarrow \dfrac{1728}{1000000}$
Now, solving the denominator part:
$\Rightarrow 0.12 \times 0.12$
$\Rightarrow \dfrac{12}{100} \times \dfrac{12}{100}$
$\Rightarrow \dfrac{12}{100} \times \dfrac{12}{100} \Rightarrow \dfrac{144}{10000}$
So, we have:
$\Rightarrow \dfrac{\dfrac{1728}{1000000}}{\dfrac{144}{10000}}$
$\Rightarrow \dfrac{1728 \times 10000}{144 \times 1000000}$
$\Rightarrow \dfrac{12}{100} \Rightarrow 0.12$ Ans
OR
$\dfrac{12}{100} = \dfrac{6}{50} = \dfrac{3}{25} = 0.12$ Ans
To solve Fractions using formula - Following identities are used to evaluate certain types of Fractions:
(a) $(a + b)^{2} = a^{2} + 2ab + b^{2}$
(b) $(a - b)^{2} = a^{2} - 2ab + b^{2}$
(c) $a^{2} - b^{2} = (a + b) (a - b)$
(d) $a^{3} + b^{3} = (a + b) (a^{2} - 2ab + b^{2})$
(e) $a^{3} - b^{3} = (a - b) (a^{2} + 2ab + b^{2})$
View More Fundamental Concepts
Q4. Evaluate using the formula:
(i) $\dfrac{1.25 \times 1.25 - 0.87 \times 0.87}{1 - 0.62}$
(ii) $\dfrac{0.87 \times 0.87 + 2 \times 0.87 \times 0.13 + 0.13 \times 0.13}{0.87 + 0.13}$
(iii) $\dfrac{2.43 \times 2.43 - 2 \times 2.43 \times 1.67 + 1.67 \times 1.67}{2.43 - 1.67}$
(iv) $\dfrac{(0.4)^{3} - (0.24)^{3}}{(0.4)^{2} + 0.4 \times 0.24 + (0.24)^{2}}$
(v) $\dfrac{0.3 \times 0.3 \times 0.3 + 0.1 \times 0.1 \times 0.1}{0.3 \times 0.3 - 0.3 \times 0.1 + 0.1 \times 0.1}$
(vi) $\dfrac{5.7 \times 5.7 - 2.3 \times 2.3}{5.7 \times 5.7 + 2 \times 5.7 \times 2.3 + 2.3 \times 2.3}$
(vii) $\dfrac{0.46 \times 0.46 + 2 \times 0.46 \times 0.14 + 0.14 \times 0.14}{0.46 \times 0.46 - 0.14 \times 0.14}$
Show Answer
(i) $\dfrac{1.25 \times 1.25 - 0.87 \times 0.87}{1 - 0.62}$
$\Rightarrow \dfrac{1.5625 - 0.7569}{1 - \dfrac{62}{100}}$
$\Rightarrow \dfrac{0.8056}{\dfrac{100 - 62}{100}}$
$\Rightarrow \dfrac{0.8056}{\dfrac{38}{100}}$
$\Rightarrow \dfrac{0.8056}{0.38}$
$\Rightarrow \dfrac{0.8056}{0.38}$
$\Rightarrow \dfrac{0.8056 \times 100}{0.38 \times 100}$
$\Rightarrow \dfrac{212}{100} \Rightarrow 2.12$ Ans
(ii) $\dfrac{0.87 \times 0.87 + 2 \times 0.87 \times 0.13 + 0.13 \times 0.13}{0.87 + 0.13}$
$\Rightarrow \text{Let a} = 0.87 \text{ and b} = 0.13$
$\Rightarrow \text{So we have: } \dfrac{a^{2} + 2ab + b^{2}}{a + b}$
$\Rightarrow \text{We know that: } (a + b)^{2} = a^{2} + 2ab + b^{2}$
$\therefore \dfrac{(a + b)^{2}}{(a + b)} \Rightarrow a + b$
$\Rightarrow a + b = 0.87 + 0.13 = 1.00 = 1$ Ans
(iii) $\dfrac{2.43 \times 2.43 - 2 \times 2.43 \times 1.67 + 1.67 \times 1.67}{2.43 - 1.67}$
$\Rightarrow$ Let $a = 2.48$ and $b = 1.67$
So we have: $\dfrac{(a \times a) - (2 \times a \times b) + (b \times b)}{a - b}$
$\Rightarrow \dfrac{a^2 - 2ab + b^2}{a - b}$
We know that: $(a - b)^2 = a^2 - 2ab + b^2$
$\therefore \dfrac{(a - b)^2}{a - b} = a - b$
$\Rightarrow 2.43 - 1.67 = 0.76$ Ans
(iv) $\dfrac{(0.4)^{3} - (0.24)^{3}}{(0.4)^{2} + 0.4 \times 0.24 + (0.24)^{2}}$
$\Rightarrow \text{Let a} = 0.4 \text{ and b} = 0.24$
$\Rightarrow \text{So we have: } \dfrac{a^{3} - b^{3}}{a^{2} + ab + b^{2}}$
$\Rightarrow \text{We know that: } a^{3} - b^{3} = (a - b)(a^{2} + ab +b^{2})$
$\therefore \dfrac{(a - b)(a^{2} + ab +b^{2})}{a^{2} + ab +b^{2}}$
$\Rightarrow a - b = 0.4 - 0.24 = \dfrac{4}{10} - \dfrac{24}{100}$
$\Rightarrow \dfrac{4}{10} - \dfrac{24}{100} \Rightarrow \dfrac{40 - 24}{100} \Rightarrow \dfrac{16}{100}$
$\Rightarrow 0.16$ Ans
(v) $\dfrac{0.3 \times 0.3 \times 0.3 + 0.1 \times 0.1 \times 0.1}{0.3 \times 0.3 - 0.3 \times 0.1 + 0.1 \times 0.1}$
$\Rightarrow \text{Let a} = 0.3 \text{ and b} = 0.1$
$\Rightarrow \text{So we have: } \dfrac{a^{3} + b^{3}}{a^{2} - ab + b^{2}}$
$\Rightarrow \text{We know that: } a^{3} + b^{3} = (a + b)(a^{2} - ab +b^{2})$
$\therefore \dfrac{(a + b)(a^{2} - ab +b^{2})}{a^{2} - ab +b^{2}}$
$\Rightarrow a + b = 0.3 + 0.1 \Rightarrow 0.1$ Ans
(vi) $\dfrac{5.7 \times 5.7 - 2.3 \times 2.3}{5.7 \times 5.7 + 2 \times 5.7 \times 2.3 + 2.3 \times 2.3}$
$\Rightarrow \text{Let a} = 5.7 \text{ and b} = 2.3$
$\Rightarrow \text{So we have: } \dfrac{a^{2} - b^{2}}{a^{2} + ab + b^{2}}$
$\Rightarrow \text{We know that: } a^{2} - b^{2} = (a + b)(a - b)$
$\Rightarrow \text{Also: } a^{2} + ab + b^{2} = (a + b)^{2}$
$\Rightarrow \text{As such, we have: } \dfrac{(a + b)(a - b)}{(a + b)^{2}} \Rightarrow \dfrac{a - b}{a + b}$
$\therefore \dfrac{(5.7 - 2.3)}{5.7 + 2.3} \Rightarrow \dfrac{3.4}{8} \Rightarrow 0.425$ Ans
(vii)$\dfrac{0.46 \times 0.46 + 2 \times 0.46 \times 0.14 + 0.14 \times 0.14}{0.46 \times 0.46 - 0.14 \times 0.14}$
$\Rightarrow \text{Let a} = 0.46 \text{ and b} = 0.14$
$\Rightarrow \text{So we have: } \dfrac{a^{2} + 2ab + b^{2}}{a^{2} - b^{2}}$
$\Rightarrow \text{We know that: } a^{2} + 2ab + b^{2} = (a + b)^{2}$
$\Rightarrow \text{Also: } a^{2} - b^{2} = (a + b)(a - b)$
$\Rightarrow \text{As such, we have: } \dfrac{(a + b)^{2}}{(a + b)(a - b)} \Rightarrow \dfrac{(a + b)(a + b)}{(a + b)(a - b)} \Rightarrow \dfrac{a + b}{a - b}$
$\therefore \dfrac{0.46 + 0.14}{0.46 - 0.14} \Rightarrow \dfrac{0.60}{0.32}$
$\Rightarrow \dfrac{0.60}{0.32} \Rightarrow \dfrac{60 \times 100}{32 \times 100} \Rightarrow \dfrac{60}{32}$
$\Rightarrow \dfrac{60}{32} \Rightarrow \dfrac{30}{16} \Rightarrow \dfrac{15}{8}$
$\Rightarrow \dfrac{15}{8} \Rightarrow 1\dfrac{1}{8} \text{ OR } 1.875$ Ans
$\Rightarrow \dfrac{1.5625 - 0.7569}{1 - \dfrac{62}{100}}$
$\Rightarrow \dfrac{0.8056}{\dfrac{100 - 62}{100}}$
$\Rightarrow \dfrac{0.8056}{\dfrac{38}{100}}$
$\Rightarrow \dfrac{0.8056}{0.38}$
$\Rightarrow \dfrac{0.8056}{0.38}$
$\Rightarrow \dfrac{0.8056 \times 100}{0.38 \times 100}$
$\Rightarrow \dfrac{212}{100} \Rightarrow 2.12$ Ans
(ii) $\dfrac{0.87 \times 0.87 + 2 \times 0.87 \times 0.13 + 0.13 \times 0.13}{0.87 + 0.13}$
$\Rightarrow \text{Let a} = 0.87 \text{ and b} = 0.13$
$\Rightarrow \text{So we have: } \dfrac{a^{2} + 2ab + b^{2}}{a + b}$
$\Rightarrow \text{We know that: } (a + b)^{2} = a^{2} + 2ab + b^{2}$
$\therefore \dfrac{(a + b)^{2}}{(a + b)} \Rightarrow a + b$
$\Rightarrow a + b = 0.87 + 0.13 = 1.00 = 1$ Ans
(iii) $\dfrac{2.43 \times 2.43 - 2 \times 2.43 \times 1.67 + 1.67 \times 1.67}{2.43 - 1.67}$
$\Rightarrow$ Let $a = 2.48$ and $b = 1.67$
So we have: $\dfrac{(a \times a) - (2 \times a \times b) + (b \times b)}{a - b}$
$\Rightarrow \dfrac{a^2 - 2ab + b^2}{a - b}$
We know that: $(a - b)^2 = a^2 - 2ab + b^2$
$\therefore \dfrac{(a - b)^2}{a - b} = a - b$
$\Rightarrow 2.43 - 1.67 = 0.76$ Ans
(iv) $\dfrac{(0.4)^{3} - (0.24)^{3}}{(0.4)^{2} + 0.4 \times 0.24 + (0.24)^{2}}$
$\Rightarrow \text{Let a} = 0.4 \text{ and b} = 0.24$
$\Rightarrow \text{So we have: } \dfrac{a^{3} - b^{3}}{a^{2} + ab + b^{2}}$
$\Rightarrow \text{We know that: } a^{3} - b^{3} = (a - b)(a^{2} + ab +b^{2})$
$\therefore \dfrac{(a - b)(a^{2} + ab +b^{2})}{a^{2} + ab +b^{2}}$
$\Rightarrow a - b = 0.4 - 0.24 = \dfrac{4}{10} - \dfrac{24}{100}$
$\Rightarrow \dfrac{4}{10} - \dfrac{24}{100} \Rightarrow \dfrac{40 - 24}{100} \Rightarrow \dfrac{16}{100}$
$\Rightarrow 0.16$ Ans
(v) $\dfrac{0.3 \times 0.3 \times 0.3 + 0.1 \times 0.1 \times 0.1}{0.3 \times 0.3 - 0.3 \times 0.1 + 0.1 \times 0.1}$
$\Rightarrow \text{Let a} = 0.3 \text{ and b} = 0.1$
$\Rightarrow \text{So we have: } \dfrac{a^{3} + b^{3}}{a^{2} - ab + b^{2}}$
$\Rightarrow \text{We know that: } a^{3} + b^{3} = (a + b)(a^{2} - ab +b^{2})$
$\therefore \dfrac{(a + b)(a^{2} - ab +b^{2})}{a^{2} - ab +b^{2}}$
$\Rightarrow a + b = 0.3 + 0.1 \Rightarrow 0.1$ Ans
(vi) $\dfrac{5.7 \times 5.7 - 2.3 \times 2.3}{5.7 \times 5.7 + 2 \times 5.7 \times 2.3 + 2.3 \times 2.3}$
$\Rightarrow \text{Let a} = 5.7 \text{ and b} = 2.3$
$\Rightarrow \text{So we have: } \dfrac{a^{2} - b^{2}}{a^{2} + ab + b^{2}}$
$\Rightarrow \text{We know that: } a^{2} - b^{2} = (a + b)(a - b)$
$\Rightarrow \text{Also: } a^{2} + ab + b^{2} = (a + b)^{2}$
$\Rightarrow \text{As such, we have: } \dfrac{(a + b)(a - b)}{(a + b)^{2}} \Rightarrow \dfrac{a - b}{a + b}$
$\therefore \dfrac{(5.7 - 2.3)}{5.7 + 2.3} \Rightarrow \dfrac{3.4}{8} \Rightarrow 0.425$ Ans
(vii)$\dfrac{0.46 \times 0.46 + 2 \times 0.46 \times 0.14 + 0.14 \times 0.14}{0.46 \times 0.46 - 0.14 \times 0.14}$
$\Rightarrow \text{Let a} = 0.46 \text{ and b} = 0.14$
$\Rightarrow \text{So we have: } \dfrac{a^{2} + 2ab + b^{2}}{a^{2} - b^{2}}$
$\Rightarrow \text{We know that: } a^{2} + 2ab + b^{2} = (a + b)^{2}$
$\Rightarrow \text{Also: } a^{2} - b^{2} = (a + b)(a - b)$
$\Rightarrow \text{As such, we have: } \dfrac{(a + b)^{2}}{(a + b)(a - b)} \Rightarrow \dfrac{(a + b)(a + b)}{(a + b)(a - b)} \Rightarrow \dfrac{a + b}{a - b}$
$\therefore \dfrac{0.46 + 0.14}{0.46 - 0.14} \Rightarrow \dfrac{0.60}{0.32}$
$\Rightarrow \dfrac{0.60}{0.32} \Rightarrow \dfrac{60 \times 100}{32 \times 100} \Rightarrow \dfrac{60}{32}$
$\Rightarrow \dfrac{60}{32} \Rightarrow \dfrac{30}{16} \Rightarrow \dfrac{15}{8}$
$\Rightarrow \dfrac{15}{8} \Rightarrow 1\dfrac{1}{8} \text{ OR } 1.875$ Ans
Q5. On a particular day, $\dfrac{2}{15}$th of the total no. of students in a school were absent. If 1950 were present on that day, find the total strenght of the school.
Show Answer
$\Rightarrow \text{Given, no. of students absent } = \dfrac{2}{15}$
$\Rightarrow \text{Let total no. of students be } = 1$
$\therefore \text{No. of students present } = 1 - \dfrac{2}{15}$
$\Rightarrow \dfrac{15 - 13}{15} \Rightarrow \dfrac{13}{15}$
$\Rightarrow \text{Given, no. of students present } = 1950$
$\Rightarrow \text{So, we have: } \dfrac{13}{15} = 1950$
$\Rightarrow \dfrac{1950 \times 15}{13}$
$\Rightarrow 150 \times 15 = 2250$ Ans
$\Rightarrow \text{Let total no. of students be } = 1$
$\therefore \text{No. of students present } = 1 - \dfrac{2}{15}$
$\Rightarrow \dfrac{15 - 13}{15} \Rightarrow \dfrac{13}{15}$
$\Rightarrow \text{Given, no. of students present } = 1950$
$\Rightarrow \text{So, we have: } \dfrac{13}{15} = 1950$
$\Rightarrow \dfrac{1950 \times 15}{13}$
$\Rightarrow 150 \times 15 = 2250$ Ans
Q6. Sangita reads $\dfrac{3}{8}$ of a book on one day and $\dfrac{4}{5}$ of the remainder on another day. Find:
(i) Portion of the book left unread after day one.
(ii) Portion of the book read on another day.
(iii) Portion of the book left unread after 2 days.
(iv) Total no. of pages in the book, if 60 pages are left unread after second day.
Show Answer
$\Rightarrow \text{Portion of book read on day one } = \dfrac{3}{8}$
$\Rightarrow \text{Let whole book be } = 1$
$\therefore \text{Portion of book left unread after day one } = 1 - \dfrac{3}{8}$
$\Rightarrow 1 - \dfrac{3}{8} \Rightarrow \dfrac{8 - 3}{8} \Rightarrow \dfrac{5}{8}$ Ans (i)
$\therefore \text{Given, portion of book read on day two } = \dfrac{4}{5} \text{ of } \dfrac{5}{8}$
$\Rightarrow \dfrac{4}{5} \times \dfrac{5}{8} \Rightarrow \dfrac{4}{8} \Rightarrow \dfrac{1}{2}$ Ans (ii)
$\therefore \text{Portion of book left unread after day two } = \dfrac{5}{8} - \dfrac{1}{2}$
$\Rightarrow \dfrac{5}{8} - \dfrac{1}{2} \Rightarrow \dfrac{5 - 4}{8} \Rightarrow \dfrac{1}{8}$ Ans (iii)
$\Rightarrow \text{Given pages left unread after day two } = 60$
$\Rightarrow \text{So, we have: } \dfrac{1}{8} = 60$
$\Rightarrow \dfrac{60 \times 8}{1} \Rightarrow 480$ Ans (iv)
$\Rightarrow \text{Let whole book be } = 1$
$\therefore \text{Portion of book left unread after day one } = 1 - \dfrac{3}{8}$
$\Rightarrow 1 - \dfrac{3}{8} \Rightarrow \dfrac{8 - 3}{8} \Rightarrow \dfrac{5}{8}$ Ans (i)
$\therefore \text{Given, portion of book read on day two } = \dfrac{4}{5} \text{ of } \dfrac{5}{8}$
$\Rightarrow \dfrac{4}{5} \times \dfrac{5}{8} \Rightarrow \dfrac{4}{8} \Rightarrow \dfrac{1}{2}$ Ans (ii)
$\therefore \text{Portion of book left unread after day two } = \dfrac{5}{8} - \dfrac{1}{2}$
$\Rightarrow \dfrac{5}{8} - \dfrac{1}{2} \Rightarrow \dfrac{5 - 4}{8} \Rightarrow \dfrac{1}{8}$ Ans (iii)
$\Rightarrow \text{Given pages left unread after day two } = 60$
$\Rightarrow \text{So, we have: } \dfrac{1}{8} = 60$
$\Rightarrow \dfrac{60 \times 8}{1} \Rightarrow 480$ Ans (iv)
Q7. Daisy spent $\dfrac{2}{5}$ of her money on food and $\dfrac{1}{3}$ on books.
Find:
(i) Fraction of money spent altogether.
(ii) Fraction of money left with Daisy.
(iii) If she had Rs.9,000, then amount left with her.
Show Answer
$\Rightarrow \text{Given, part spent on food } = \dfrac{2}{5}$
$\Rightarrow \text{Given, part spent on books } = \dfrac{1}{3}$
$\therefore \text{Fraction of money spent altogether } = \dfrac{2}{5} + \dfrac{1}{3}$
$\Rightarrow \dfrac{2}{5} + \dfrac{1}{3} \Rightarrow \dfrac{6 + 5}{15} \Rightarrow \dfrac{11}{15}$ Ans (i)
$\therefore \text{Fraction of money left with Daisy } = 1 - \dfrac{11}{15}$
$\Rightarrow 1 - \dfrac{11}{15} \Rightarrow \dfrac{15 - 11}{15} \Rightarrow \dfrac{4}{15}$ Ans (ii)
$\therefore \text{Amount of money left with Daisy }= \dfrac{4}{15} \text{ of } 9000$
$\Rightarrow \dfrac{4}{15} \times 9000 \Rightarrow 4 \times 600 \Rightarrow \text{Rs.}2400$ Ans (iii)
$\Rightarrow \text{Given, part spent on books } = \dfrac{1}{3}$
$\therefore \text{Fraction of money spent altogether } = \dfrac{2}{5} + \dfrac{1}{3}$
$\Rightarrow \dfrac{2}{5} + \dfrac{1}{3} \Rightarrow \dfrac{6 + 5}{15} \Rightarrow \dfrac{11}{15}$ Ans (i)
$\therefore \text{Fraction of money left with Daisy } = 1 - \dfrac{11}{15}$
$\Rightarrow 1 - \dfrac{11}{15} \Rightarrow \dfrac{15 - 11}{15} \Rightarrow \dfrac{4}{15}$ Ans (ii)
$\therefore \text{Amount of money left with Daisy }= \dfrac{4}{15} \text{ of } 9000$
$\Rightarrow \dfrac{4}{15} \times 9000 \Rightarrow 4 \times 600 \Rightarrow \text{Rs.}2400$ Ans (iii)
Q8. A man leaves behind Rs.1,15,500. $\dfrac{1}{3}$ was to be recieved by his wife, $\dfrac{2}{7}$ to each of his two daughters and the rest to his son.
Find the amount recieved by his son.
Show Answer
$\Rightarrow \text{Given, Total money left } = 1,15,500$
$\therefore \text{Part of wife } = \dfrac{1}{3} \text{ of } 1,15,500$
$\Rightarrow \dfrac{1}{3} \times 1,15,500 \Rightarrow \text{ Rs.}38,500$
$\Rightarrow \text{Reamining amount } = 1,15,500 - 38,500 \Rightarrow 77,000$
$\therefore \text{Part of one daughter } = \dfrac{2}{7} \text{ of } 77,000$
$\Rightarrow \dfrac{2}{7} \times 77,000 \Rightarrow \text{ Rs.}22,000$
$\therefore \text{Part of two daughters } = 22,000 \times 2 \Rightarrow \text{ Rs.}44,000$
$\Rightarrow \text{Reamining amount } = 77,000 - 44,000 \Rightarrow \text{ Rs.}33,000$
$\therefore \text{Part of son } = \text{Rs.}33,000$ Ans
$\therefore \text{Part of wife } = \dfrac{1}{3} \text{ of } 1,15,500$
$\Rightarrow \dfrac{1}{3} \times 1,15,500 \Rightarrow \text{ Rs.}38,500$
$\Rightarrow \text{Reamining amount } = 1,15,500 - 38,500 \Rightarrow 77,000$
$\therefore \text{Part of one daughter } = \dfrac{2}{7} \text{ of } 77,000$
$\Rightarrow \dfrac{2}{7} \times 77,000 \Rightarrow \text{ Rs.}22,000$
$\therefore \text{Part of two daughters } = 22,000 \times 2 \Rightarrow \text{ Rs.}44,000$
$\Rightarrow \text{Reamining amount } = 77,000 - 44,000 \Rightarrow \text{ Rs.}33,000$
$\therefore \text{Part of son } = \text{Rs.}33,000$ Ans
Q9. Evaluate:
(i) $0.4 \text{ of Rs.}55 + \dfrac{1}{7} \text{ of Rs.}42$
(ii) $\dfrac{5}{9} \text{ of } 810 \text{ gm } - 0.05 \text{ times of }2 \text{ kg }$
(iii) $\dfrac{2}{3} \text{ of }72 \text{ cm }+ 0.75 \text{ of } 2 \text{m } - 0.4 \text{ of } 3 \text{m }$
Show Answer
(i) $(\dfrac{4}{10} \times 55) + (\dfrac{1}{7} \times 42)$
$\Rightarrow (\dfrac{2}{5} \times 55) + 6$
$\Rightarrow (2 \times 11) + 6$
$\Rightarrow 22 + 6 = \text{ Rs.}28$ Ans
(ii) $\because 1 \text{ kg }= 1000 \text{ grams }$
$\therefore 2 \text{ kg }= 2000 \text{ grams }$
As such, we have:
$\Rightarrow (\dfrac{5}{9} \times 810) - (\dfrac{5}{100} \times 2000)$
$\Rightarrow (5 \times 90) - (5 \times 20)$
$\Rightarrow 450 - 100 = 350 \text{ grams}$ Ans
(iii) $(\dfrac{2}{3} \times 72) + (\dfrac{75}{100} \times 200) - (\dfrac{4}{10} \times 300)$
$\Rightarrow (2 \times 24) + (75 \times 2) - (4 \times 30)$
$\Rightarrow 48 + 150 - 120$
$\Rightarrow 198 - 120 = 78$ Ans
$\Rightarrow (\dfrac{2}{5} \times 55) + 6$
$\Rightarrow (2 \times 11) + 6$
$\Rightarrow 22 + 6 = \text{ Rs.}28$ Ans
(ii) $\because 1 \text{ kg }= 1000 \text{ grams }$
$\therefore 2 \text{ kg }= 2000 \text{ grams }$
As such, we have:
$\Rightarrow (\dfrac{5}{9} \times 810) - (\dfrac{5}{100} \times 2000)$
$\Rightarrow (5 \times 90) - (5 \times 20)$
$\Rightarrow 450 - 100 = 350 \text{ grams}$ Ans
(iii) $(\dfrac{2}{3} \times 72) + (\dfrac{75}{100} \times 200) - (\dfrac{4}{10} \times 300)$
$\Rightarrow (2 \times 24) + (75 \times 2) - (4 \times 30)$
$\Rightarrow 48 + 150 - 120$
$\Rightarrow 198 - 120 = 78$ Ans