Question 1: Add

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

Answer:

 $\displaystyle \text{ i) } \frac{16 }{ 21 } + \frac{23 }{ 28 }$$\displaystyle \text{ i) } \frac{16 }{ 21 } + \frac{23 }{ 28 }$ LCM of $21$$21$ and $28 = 84$$28 = 84$ $\displaystyle = \frac{64 }{ 84 } + \frac{69 }{ 84 } = \frac{133 }{ 84 }$$\displaystyle = \frac{64 }{ 84 } + \frac{69 }{ 84 } = \frac{133 }{ 84 }$ $\displaystyle \text{ ii) } 6 \frac{7 }{ 12 } + 4 \frac{7 }{ 18 } = \frac{79 }{ 12 } + \frac{79 }{ 18 }$$\displaystyle \text{ ii) } 6 \frac{7 }{ 12 } + 4 \frac{7 }{ 18 } = \frac{79 }{ 12 } + \frac{79 }{ 18 }$ LCM of $\displaystyle 12$$\displaystyle 12$ and $\displaystyle 18 = 36$$\displaystyle 18 = 36$ $\displaystyle = \frac{237 }{ 36} + \frac{158 }{ 36 } = \frac{395 }{ 36 }$$\displaystyle = \frac{237 }{ 36} + \frac{158 }{ 36 } = \frac{395 }{ 36 }$ $\displaystyle \text{ iii) } 9 \frac{3 }{ 4 } + 7 \frac{7 }{ 8 } +3 \frac{5 }{ 12 } = \frac{39 }{ 4 } + \frac{63 }{ 8 } + \frac{41 }{ 12 }$$\displaystyle \text{ iii) } 9 \frac{3 }{ 4 } + 7 \frac{7 }{ 8 } +3 \frac{5 }{ 12 } = \frac{39 }{ 4 } + \frac{63 }{ 8 } + \frac{41 }{ 12 }$ LCM of  $\displaystyle 4, 8, 12 = 24$$\displaystyle 4, 8, 12 = 24$ $\displaystyle = \frac{156 }{ 24 } + \frac{189 }{ 24 } + \frac{82 }{ 24 } = \frac{427 }{ 24 }$$\displaystyle = \frac{156 }{ 24 } + \frac{189 }{ 24 } + \frac{82 }{ 24 } = \frac{427 }{ 24 }$

$\\$

Question 2: Subtract

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

Answer:

$\displaystyle \text{ i) } \frac{9 }{ 10} - \frac{7 }{ 15}$

LCM of $\displaystyle 10$ and $\displaystyle 15 = 30$

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

$\displaystyle \text{ ii) } 13-6 \frac{2 }{ 5}$

$\displaystyle =13- \frac{32 }{ 5}$

LCM of $\displaystyle 1$ and $\displaystyle 5 = 5$

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

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

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

LCM of $\displaystyle 35$ and $\displaystyle 30 = 210$

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

$\displaystyle \\$

Question 3: Evaluate

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

Answer:

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

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

LCM of $\displaystyle 4, 6$ and $\displaystyle 8 = 24$

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

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

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

LCM of $\displaystyle 18, 12$ and $\displaystyle 24 = 144$

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

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

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

LCM of $\displaystyle 14, 21$ and $\displaystyle 42 = 42$

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

$\displaystyle \\$

Question 4: Evaluate:

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

Answer:

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

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

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

$\displaystyle \\$

Question 5: Evaluate

$\displaystyle \text{ ii) } 18 \div 2 \frac{2 }{ 3}$      $\displaystyle \text{ ii) } \frac{2 }{ 3} \div 3 \frac{3 }{ 4}$           $\displaystyle \text{ iii) } 4 \frac{2 }{ 3} \div 7$

Answer:

$\displaystyle \text{ ii) } 18 \div 2 \frac{2 }{ 3} = \frac{18 }{ 1} \times \frac{3 }{ 8} = \frac{27 }{ 4}$

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

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

$\displaystyle \\$

Question 6: Simplify:

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

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

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

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

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

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

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

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

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

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

Answer:

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle = \frac{1}{18}$

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

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

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

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

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

$\displaystyle = 25$

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

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

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

$\displaystyle = 5- \frac{229}{84}$

$\displaystyle = \frac{191}{84}$

$\displaystyle = 2 \frac{23}{84}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle = \frac{1}{6}$

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

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

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

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

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

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

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

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

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

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

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

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

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