Question 1: Add

i) \frac{16 }{ 21 } + \frac{23 }{ 28 }       ii)     6 \frac{7 }{ 12 } + 4 \frac{7 }{ 18 }     iii)    9 \frac{3 }{ 4 } + 7 \frac{7 }{ 8 } +3 \frac{5 }{ 12 }

Answer:

i) \frac{16 }{ 21 } + \frac{23 }{ 28 }

LCM of 21 and 28 = 84

= \frac{64 }{ 84 } + \frac{69 }{ 84 } = \frac{133 }{ 84 }

 

ii) 6 \frac{7 }{ 12 } + 4 \frac{7 }{ 18 }

= \frac{79 }{ 12 } + \frac{79 }{ 18 }

LCM of 12 and 18 = 36

= \frac{237 }{ 36} + \frac{158 }{ 36 } = \frac{395 }{ 36 }

iii) 9 \frac{3 }{ 4 } + 7 \frac{7 }{ 8 } +3 \frac{5 }{ 12 }

= \frac{39 }{ 4 } + \frac{63 }{ 8 } + \frac{41 }{ 12 }

LCM of 4, 8, 12 = 24

= \frac{156 }{ 24 } + \frac{189 }{ 24 } + \frac{82 }{ 24 } = \frac{427 }{ 24 }

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Question 2: Subtract

i) \frac{9 }{ 10} - \frac{7 }{ 15}       ii) 13-6 \frac{ 2 }{ 5}       iii) 2 \frac{13 }{ 35} - 1 \frac{7 }{ 30}

Answer:

i) \frac{9 }{ 10} - \frac{7 }{ 15}

LCM of 10 and 15 = 30

=\frac{27 }{ 30} - \frac{14 }{ 30} = \frac{13 }{ 30}

ii) 13-6 \frac{2 }{ 5}

=13- \frac{32 }{ 5}

LCM of 1 and 5 = 5

= \frac{65 }{ 5} - \frac{32 }{ 5} = \frac{33 }{ 5 }

iii) 2 \frac{13 }{ 35} - 1 \frac{7 }{ 30 }

= \frac{83 }{ 35} - \frac{37 }{ 30 }

LCM of 35 and 30 = 210

= \frac{498 }{ 210} - \frac{259 }{ 210} = \frac{239 }{ 210}

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Question 3: Evaluate

i) 3 \frac{1 }{ 4} - 1 \frac{5 }{ 6} +2 \frac{ 3 }{ 8}             ii) 2 \frac{1 }{ 18} - \frac{7 }{ 12} - \frac{23 }{ 24}              iii) 9 \frac{5 }{ 14} - 6 \frac{8 }{ 21} + \frac{25 }{ 42}

Answer:

i) 3 \frac{1 }{ 4} - 1 \frac{5 }{ 6} +2 \frac{ 3 }{ 8}

= \frac{13 }{ 4} - \frac{11 }{ 6} + \frac{19 }{ 8}

LCM of 4, 6 and 8 = 24

= \frac{(78-44+57) }{ 24 }   = \frac{91 }{ 24 }

 

ii) 2 \frac{1 }{ 18} - \frac{7 }{ 12} - \frac{23 }{ 24}

= \frac{37 }{ 18} -  \frac{7 }{ 12} - \frac{23 }{ 24}

LCM of 18, 12 and 24 = 144

= \frac{(296-84-138) }{ 144}   = \frac{74 }{ 144} = \frac{37 }{ 72 }

iii) 9 \frac{5 }{ 14} - 6 \frac{8 }{ 21} + \frac{25 }{ 42}

= \frac{131 }{ 14} - \frac{134 }{ 21} + \frac{25 }{ 42}

LCM of 14, 21 and 42 = 42

= \frac{(393-268+25) }{ 42}   = \frac{150 }{ 42} = \frac{75 }{ 21} = \frac{25 }{ 7}

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Question 4: Evaluate:

i) \frac{9 }{ 14} \times \frac{7 }{ 3}            ii) 6 \frac{2 }{ 3} \times 3 \frac{3 }{ 4}            iii) 3 \frac{1 }{ 3} \times \frac{18 }{ 25}

Answer:

i) \frac{9 }{ 14} \times \frac{7 }{ 3} = \frac{ (3 \times 3 \times 7) }{ (2 \times 7 \times 3)} = \frac{3 }{ 2 }

ii) 6 \frac{2 }{ 3} \times 3 \frac{3 }{ 4} = \frac{20 }{ 3} \times \frac{15 }{ 4} = \frac{(4 \times 5 \times 3 \times 5) }{ (3 \times 4)} =25

iii) 3 \frac{1 }{ 3} \times \frac{18 }{ 25} = \frac{10 }{ 3} \times 1 \frac{8 }{ 25} = \frac{(2 \times 5 \times 3 \times 6) }{ (3 \times 5 \times 5)} = \frac{12 }{ 5}

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Question 5: Evaluate

i) 18 \div 2 \frac{2 }{ 3}       ii) \frac{2 }{ 3} \div 3 \frac{3 }{ 4}           iii) 4 \frac{2 }{ 3} \div 7

Answer:

i) 18 \div 2 \frac{2 }{ 3} = \frac{18 }{ 1} \times \frac{3 }{ 8} = \frac{27 }{ 4}

ii) 11 \frac{2 }{ 3} \div 3 \frac{3 }{ 4} = \frac{35 }{ 3} \times \frac{4 }{ 15} = \frac{(7 \times 5 \times 4) }{ (3 \times 3 \times 5)} = \frac{28 }{ 9}

iii) 4 \frac{2 }{ 3} \div 7 = \frac{14 }{ 3} \times \frac{1 }{ 7} = \frac{(2 \times 7 \times 1) }{ (3 \times 7)} = \frac{2 }{ 3}

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Question 6: Simplify:

i) 1- 2 \frac{2}{5} \div 4 \frac{1}{2} \ of \ 2 \frac{2}{3} \times \frac{5}{6} + \frac{1}{3}                         ii) 1 \div \frac{5}{7} \ of \ 6 \frac{3}{10} - \frac{1}{6}

iii) \Big( \frac{1}{5} \div \frac{1}{5} \ of  \ \frac{1}{5} \Big) \div \Big( \frac{1}{5} \ of \ \frac{1}{5} \div \frac{1}{5} \Big)              iv) 5 - \Big[ \frac{3}{4} + \Big\{ 2 \frac{1}{2} - \Big( \frac{1}{2} + \frac{1}{6} - \frac{1}{7} \Big) \Big\} \Big]

v) \frac{1}{5} \Big[ 5- \frac{1}{5} \Big\{ 5- \frac{1}{5} \Big(5- \frac{1}{5} \Big) \Big\} \Big] \div 1 \frac{1}{5}                vi)    1+1 \div \Big\{1+1 \div \Big( 1+ \frac{1}{3} \Big) \Big\}

vii) 7 \frac{1}{2} - \Big[ 2 \frac{1}{4} \div \Big\{1 \frac{1}{4} - \frac{1}{2} \Big( 1 \frac{1}{2} - \frac{1}{3} - \frac{1}{6} \Big) \Big\} \Big]      viii) 3 \frac{1}{3} \div 2 \frac{1}{2} \times \frac{3}{4} \div \frac{1}{3} \ of \  21 \times 1 \frac{1}{6}

ix)   \frac{3}{4} \div 2 \frac{1}{4} \ of \   \frac{2}{3} - \Big( \frac{1}{2} - \frac{1}{3} \Big) \div \Big( \frac{1}{2} + \frac{1}{3} \Big) \times 3 \frac{1}{3} + \frac{5}{6}      

x) \Big( 3 \frac{1}{4} - \frac{4}{5} \ of \ \frac{5}{6} \Big) \div \Big\{ 4 \frac{1}{3} \div \frac{1}{5} - \Big( \frac{3}{10} +21 \frac{1}{5} \Big) \Big\}

Answer:

i)       1- 2 \frac{2}{5} \div 4 \frac{1}{2} \ of \ 2 \frac{2}{3} \times \frac{5}{6} + \frac{1}{3}

= 1- \frac{12}{5} \div \frac{9}{2} of \frac{8}{3} \times \frac{5}{6} + \frac{1}{3}

= 1- \frac{12}{5} \div 12 \times \frac{5}{6} + \frac{1}{3}

= 1- \frac{12}{5} \times \frac{1}{12} \times \frac{5}{6} + \frac{1}{3}

= 1- \frac{1}{6} + \frac{1}{3}

= \frac{7}{6} =1 \frac{1}{6}

ii)       1 \div \frac{5}{7} \ of \ 6 \frac{3}{10} - \frac{1}{6}

= 1 \div \frac{5}{7} of \frac{63}{10} - \frac{1}{6}

= 1 \div \frac{9}{2} - \frac{1}{6}

= 1 \times \frac{2}{9} - \frac{1}{6}

= \frac{(4-3)}{18}

= \frac{1}{18}

iii)      \Big( \frac{1}{5} \div \frac{1}{5} \ of  \ \frac{1}{5} \Big) \div \Big( \frac{1}{5} \ of \ \frac{1}{5} \div \frac{1}{5} \Big)

= \Big( \frac{1}{5} \div \frac{1}{25} \Big) \div \Big( \frac{1}{25} \div \frac{1}{5} \Big)

= \Big( \frac{1}{5} \times \frac{25}{1} \Big) \div \Big( \frac{1}{25} \times \frac{5}{1} \Big)

= \Big( \frac{5}{1} \Big) \div \Big( \frac{1}{5} \Big)

= \Big( \frac{5}{1} \Big) \times \Big( \frac{5}{1} \Big)

= 25

iv)     5 - \Big[ \frac{3}{4} + \Big\{ 2 \frac{1}{2} - \Big( \frac{1}{2} + \frac{1}{6} - \frac{1}{7} \Big) \Big\} \Big]

= 5-\Big[ \frac{3}{4} +\Big\{ \frac{5}{2} - \Big( \frac{1}{2} + \frac{1}{42} \Big) \Big\} \Big]

= 5-\Big[ \frac{3}{4} + \Big\{ \frac{5}{2} - \frac{22}{42}    \Big\}  \Big]

= 5- \frac{229}{84}

= \frac{191}{84}

= 2 \frac{23}{84}

v)      \frac{1}{5} \Big[ 5- \frac{1}{5} \Big\{ 5- \frac{1}{5} \Big(5- \frac{1}{5} \Big) \Big\} \Big] \div 1 \frac{1}{5}

= \frac{1}{5} \Big[5- \frac{1}{5}   \Big\{5- \frac{1}{5} \Big(5- \frac{1}{5} \Big) \Big\} \Big] \div \frac{6}{5}

= \frac{1}{5} \Big[ 5 - \frac{1}{5} \Big\{ 5 - \frac{1}{5} \Big( \frac{24}{5} \Big) \Big\} \Big] \div \frac{6}{5}

= \frac{1}{5} \Big[ 5 - \frac{1}{5} \Big\{ 5 - \frac{24}{25} \Big\} \Big] \div \frac{6}{5}

= \Big[5- \frac{1}{5} \Big\{ \frac{101}{25} \Big\} \Big] \div \frac{6}{5}

= \Big[5- \frac{101}{125} \Big] \div \frac{6}{5}

= \Big [ \frac{524}{125} \Big] \times \frac{5}{6}

= \frac{524}{125} \times \frac{5}{6}

= \frac{262}{75} =3 \frac{37}{75}

vi)    1+1 \div \Big\{1+1 \div \Big( 1+ \frac{1}{3} \Big) \Big\}

= 1+1 \div \Big\{1+1 \div \Big( \frac{4}{3} \Big) \Big\}

= 1+1 \div \Big\{1+1 \times \frac{3}{4} \Big\}

= 1+1 \div \frac{7}{4}

= 1+1 \times \frac{4}{7} = \frac{11}{7}

vii)    7 \frac{1}{2} - \Big[ 2 \frac{1}{4} \div \Big\{1 \frac{1}{4} - \frac{1}{2} \Big( 1 \frac{1}{2} - \frac{1}{3} - \frac{1}{6} \Big) \Big\} \Big]

= \frac{15}{2} - \Big[ \frac{9}{4} \div \Big\{ \frac{5}{4} -\frac{1}{2} \Big( \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \Big) \Big\} \Big]

= \frac{15}{2} - \Big[ \frac{9}{4} \div \Big\{ \frac{5}{4} - \frac{1}{2} \Big\} \Big]

= \frac{15}{2} - \Big[ \frac{9}{4} \div \frac{1}{4} \Big]

= \frac{15}{2} - \Big[ \frac{9}{4} \times \frac{4}{1} \Big]

= \frac{15}{2} -9 = \frac{-3}{2}

viii)  3 \frac{1}{3} \div 2 \frac{1}{2} \times \frac{3}{4} \div \frac{1}{3} \ of \  21 \times 1 \frac{1}{6}

= \frac{10}{3} \div \frac{5}{2} \times \frac{3}{4} \div \frac{1}{3} \ of \  21 \times \frac{7}{6}

= \frac{10}{3} \div \frac{5}{2} \times \frac{3}{4} \div 7 \times \frac{7}{6}

= \frac{10}{3} \times \frac{2}{5} \times \frac{3}{4} \times \frac{1}{7} \times \frac{7}{6}

= \frac{1}{6}

ix)   \frac{3}{4} \div 2 \frac{1}{4} \ of \   \frac{2}{3} - \Big( \frac{1}{2} - \frac{1}{3} \Big) \div \Big( \frac{1}{2} + \frac{1}{3} \Big) \times 3 \frac{1}{3} + \frac{5}{6}

= \frac{3}{4} \div \frac{9}{4} \ of \  \frac{2}{3} - \Big( \frac{1}{2} - \frac{1}{3} \Big) \div \Big( \frac{1}{2} + \frac{1}{3} \Big) \times \frac{10}{3} + \frac{5}{6}

= \frac{3}{4} \div \frac{9}{4} \ of \  \frac{2}{3} - \Big( \frac{1}{6} \Big) \div \Big( \frac{5}{6} \Big) \times \frac{10}{3} + \frac{5}{6}

= \frac{3}{4} \div \frac{9}{4} \ of \  \frac{2}{3} - \frac{1}{6} \times \frac{6}{5} \times \frac{10}{3} + \frac{5}{6}

= \frac{3}{4} \div \frac{9}{4} \ of \  \frac{2}{3} - \frac{2}{3} + \frac{5}{6}

= \frac{3}{4} \div \frac{3}{2} - \frac{2}{3} + \frac{5}{6}

= \frac{3}{4} \times \frac{2}{3} - \frac{2}{3} + \frac{5}{6}

= \frac{1}{2} - \frac{2}{3} + \frac{5}{6} = \frac{2}{3}

x)      \Big( 3 \frac{1}{4} - \frac{4}{5} \ of \ \frac{5}{6} \Big) \div \Big\{ 4 \frac{1}{3} \div \frac{1}{5} - \Big( \frac{3}{10} +21 \frac{1}{5} \Big) \Big\}

= \Big( \frac{13}{4} - \frac{4}{5} \ of \ \frac{5}{6} \Big) \div \Big\{ \frac{13}{3} \div \frac{1}{5} - \Big( \frac{3}{10} + \frac{106}{5} \Big) \Big\}

= \Big( \frac{13}{4} - \frac{4}{5} \ of \ \frac{5}{6} \Big) \div \Big\{ \frac{13}{3} \times \frac{5}{1} - \frac{43}{2} \Big\}

= \Big( \frac{13}{4} - \frac{2}{3} \Big) \div \Big\{ \frac{65}{3} - \frac{43}{2} \Big\}

= \frac{31}{2} \times \frac{6}{1} = 93