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

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

$\\$

Question 2: Subtract

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

 i) $\frac{9 }{ 10}$$\frac{9 }{ 10}$ $-$$-$ $\frac{7 }{ 15}$$\frac{7 }{ 15}$ LCM of $10$$10$ and $15 = 30$$15 = 30$ $=\frac{27 }{ 30}$$=\frac{27 }{ 30}$ $-$$-$ $\frac{14 }{ 30}$$\frac{14 }{ 30}$ $=$$=$ $\frac{13 }{ 30}$$\frac{13 }{ 30}$ ii) $13-6$$13-6$ $\frac{2 }{ 5}$$\frac{2 }{ 5}$ $=13-$$=13-$ $\frac{32 }{ 5}$$\frac{32 }{ 5}$ LCM of $1$$1$ and $5 = 5$$5 = 5$ $=$$=$ $\frac{65 }{ 5}$$\frac{65 }{ 5}$ $-$$-$ $\frac{32 }{ 5}$$\frac{32 }{ 5}$ $=$$=$ $\frac{33 }{ 5 }$$\frac{33 }{ 5 }$ iii) $2$$2$ $\frac{13 }{ 35}$$\frac{13 }{ 35}$ $- 1$$- 1$ $\frac{7 }{ 30 }$$\frac{7 }{ 30 }$ $=$$=$ $\frac{83 }{ 35}$$\frac{83 }{ 35}$ $-$$-$ $\frac{37 }{ 30 }$$\frac{37 }{ 30 }$ LCM of $35$$35$ and $30 = 210$$30 = 210$ $=$$=$ $\frac{498 }{ 210}$$\frac{498 }{ 210}$ $-$$-$ $\frac{259 }{ 210}$$\frac{259 }{ 210}$ $=$$=$ $\frac{239 }{ 210}$$\frac{239 }{ 210}$

$\\$

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

 i) $3$$3$ $\frac{1 }{ 4}$$\frac{1 }{ 4}$ $- 1$$- 1$ $\frac{5 }{ 6}$$\frac{5 }{ 6}$ $+2$$+2$ $\frac{ 3 }{ 8}$$\frac{ 3 }{ 8}$ $=$$=$ $\frac{13 }{ 4}$$\frac{13 }{ 4}$ $-$$-$ $\frac{11 }{ 6}$$\frac{11 }{ 6}$ $+$$+$ $\frac{19 }{ 8}$$\frac{19 }{ 8}$ LCM of $4, 6$$4, 6$ and $8 = 24$$8 = 24$ $=$$=$ $\frac{(78-44+57) }{ 24 }$$\frac{(78-44+57) }{ 24 }$  $=$$=$ $\frac{91 }{ 24 }$$\frac{91 }{ 24 }$ ii) $2$$2$ $\frac{1 }{ 18}$$\frac{1 }{ 18}$ $-$$-$ $\frac{7 }{ 12}$$\frac{7 }{ 12}$ $-$$-$ $\frac{23 }{ 24}$$\frac{23 }{ 24}$ $=$$=$ $\frac{37 }{ 18}$$\frac{37 }{ 18}$ $-$$-$ $\frac{7 }{ 12}$$\frac{7 }{ 12}$ $-$$-$ $\frac{23 }{ 24}$$\frac{23 }{ 24}$ LCM of $18, 12$$18, 12$ and $24 = 144$$24 = 144$ $=$$=$ $\frac{(296-84-138) }{ 144}$$\frac{(296-84-138) }{ 144}$  $=$$=$ $\frac{74 }{ 144}$$\frac{74 }{ 144}$ $=$$=$ $\frac{37 }{ 72 }$$\frac{37 }{ 72 }$ iii) $9$$9$ $\frac{5 }{ 14}$$\frac{5 }{ 14}$ $- 6$$- 6$ $\frac{8 }{ 21}$$\frac{8 }{ 21}$ $+$$+$ $\frac{25 }{ 42}$$\frac{25 }{ 42}$ $=$$=$ $\frac{131 }{ 14}$$\frac{131 }{ 14}$ $-$$-$ $\frac{134 }{ 21}$$\frac{134 }{ 21}$ $+$$+$ $\frac{25 }{ 42}$$\frac{25 }{ 42}$ LCM of $14, 21$$14, 21$ and $42 = 42$$42 = 42$ $=$$=$ $\frac{(393-268+25) }{ 42}$$\frac{(393-268+25) }{ 42}$  $=$$=$ $\frac{150 }{ 42}$$\frac{150 }{ 42}$ $=$$=$ $\frac{75 }{ 21}$$\frac{75 }{ 21}$ $=$$=$ $\frac{25 }{ 7}$$\frac{25 }{ 7}$

$\\$

Question 4: Evaluate:

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

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

$\\$

Question 5: Evaluate

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

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

$\\$

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

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 \$ $\frac{5}{6}$ $\Big) \div \Big\{$ $\frac{13}{3}$ $\times$ $\frac{5}{1}$ $-$ $\frac{43}{2}$ $\Big\}$

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

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