Question 1: Express each of the following as an improper fractions:

$\displaystyle \text{ i) } 9 \frac{14}{15}$      $\displaystyle \text{ii) }17 \frac{5}{6}$      $\displaystyle \text{iii) }13 \frac{11}{26}$       $\displaystyle \text{ iv) } 3 \frac{41}{51}$

$\displaystyle \text{ i) } 9 \frac{14}{15} = \frac{(9 \times 15+14)}{15} = \frac{149}{15}$

$\displaystyle \text{ii) } 17 \frac{5}{6} = \frac{(17 \times 6+5)}{6} = \frac{107}{6}$

$\displaystyle \text{ iii) } 13 \frac{11}{26} = \frac{(13 \times 26+11)}{26} = \frac{349}{26}$

$\displaystyle \text{ iv) } 3 \frac{41}{51} = \frac{(3 \times 51+41)}{51} = \frac{194}{51}$

$\displaystyle \\$

Question 2:

$\displaystyle \text{ i) } \frac{135}{26}$     $\displaystyle \text{ ii) } \frac{148}{35}$     $\displaystyle \text{ iii) } \frac{620}{17}$     $\displaystyle \text{ iv) } \frac{1075}{96}$

 i) $\displaystyle \frac{135}{26} = 5 \frac{5}{26}$$\displaystyle \frac{135}{26} = 5 \frac{5}{26}$   $\underline{\hspace{0.5cm}26} ) \overline{\hspace{0.5cm} 135 \hspace{0.5cm} } (\underline{5\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 130} \\ {\hspace{1.7cm} 5}$$\underline{\hspace{0.5cm}26} ) \overline{\hspace{0.5cm} 135 \hspace{0.5cm} } (\underline{5\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 130} \\ {\hspace{1.7cm} 5}$ ii) $\displaystyle \frac{148}{35} = 4 \frac{8}{35}$$\displaystyle \frac{148}{35} = 4 \frac{8}{35}$   $\underline{\hspace{0.5cm}35} ) \overline{\hspace{0.5cm} 148 \hspace{0.5cm} } (\underline{4\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 140} \\ {\hspace{1.7cm} 8}$$\underline{\hspace{0.5cm}35} ) \overline{\hspace{0.5cm} 148 \hspace{0.5cm} } (\underline{4\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 140} \\ {\hspace{1.7cm} 8}$ iii) $\displaystyle \frac{620}{17} = 36 \frac{8}{17}$$\displaystyle \frac{620}{17} = 36 \frac{8}{17}$   $\underline{\hspace{0.5cm}17} ) \overline{\hspace{0.5cm} 620 \hspace{0.5cm} } (\underline{36\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 612} \\ {\hspace{1.7cm} 8}$$\underline{\hspace{0.5cm}17} ) \overline{\hspace{0.5cm} 620 \hspace{0.5cm} } (\underline{36\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 612} \\ {\hspace{1.7cm} 8}$ iv) $\displaystyle \frac{1075}{96} = 11 \frac{19}{96}$$\displaystyle \frac{1075}{96} = 11 \frac{19}{96}$   $\underline{\hspace{0.5cm}96} ) \overline{\hspace{0.5cm} 1075 \hspace{0.5cm} } (\underline{11\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 1056} \\ {\hspace{1.7cm} 19}$$\underline{\hspace{0.5cm}96} ) \overline{\hspace{0.5cm} 1075 \hspace{0.5cm} } (\underline{11\hspace{0.5cm}} \\ \underline{\hspace{1.5cm} 1056} \\ {\hspace{1.7cm} 19}$

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Question 3: Write five fractions equivalent to each of the following fractions:

$\displaystyle \text{ i) } \frac{4}{7}$     $\displaystyle \text{ ii) }\frac{128 }{ 192}$

$\displaystyle \text{ i) } \frac{4}{7} = \frac{8 }{ 14} = \frac{12 }{ 21} = \frac{16 }{ 28} = \frac{20 }{ 35} = \frac{24 }{ 42}$

$\displaystyle \text{ ii) } \frac{128 }{ 192} = \frac{256 }{ 384} = \frac{384 }{ 576} = \frac{512 }{ 768} = \frac{640 }{ 960} = \frac{768 }{ 1152}$

$\displaystyle \\$

Question 4: Convert the unlike fractions into like fractions:

$\displaystyle \text{ i) } \frac{3 }{ 4} , \frac{5 }{ 8} , \frac{7 }{ 12} , \frac{13 }{ 24}$       $\displaystyle \text{ ii) } \frac{1 }{ 3} , \frac{4 }{ 5} , \frac{7 }{ 10} , \frac{11 }{ 15}$

$\displaystyle \text{ i) } \frac{3 }{ 4} , \frac{5 }{ 8} , \frac{7 }{ 12} , \frac{13 }{ 24}$

$\displaystyle \text{ L.C.M of } 4, 8, 12, 24 = 24$.

$\displaystyle \text{ Therefore the fractions can be written as following like fractions } \frac{18 }{ 24} , \frac{15 }{ 24} , \frac{14 }{ 24} , \frac{13 }{ 24}$

$\displaystyle \text{ ii) }\frac{1 }{ 3} , \frac{4 }{ 5} , \frac{7 }{ 10} , \frac{11 }{ 15}$

$\displaystyle \text{ L.C.M of } 3, 5, 10, 15 = 30$.

$\displaystyle \text{ Therefore the fractions can be written as following like fractions } \frac{10 }{ 30} , \frac{24 }{ 30} , \frac{21 }{ 30} , \frac{22 }{ 30}$

$\displaystyle \\$

Question 5: Reduce each of the following into the simplest form:

i) $\displaystyle \frac{153}{221}$        ii) $\displaystyle \frac{115}{138}$         iii) $\displaystyle \frac{87}{116}$

$\displaystyle \text{ i) } \frac{153}{221} = \frac{3 \times 3 \times 17}{17 \times 13} = \frac{9}{13}$

 $\underline{3 \ \ \ } | \underline{153} \\ \underline{3 \ \ \ } | \underline{51} \\ \underline{17 \ } | \underline{17} \\ \underline{\ \ \ \ } | \underline{1} \\$$\underline{3 \ \ \ } | \underline{153} \\ \underline{3 \ \ \ } | \underline{51} \\ \underline{17 \ } | \underline{17} \\ \underline{\ \ \ \ } | \underline{1} \\$ $\underline{17 \ } | \underline{221} \\ \underline{3 \ \ \ } | \underline{51} \\ \underline{\ \ \ \ } | \underline{1} \\$$\underline{17 \ } | \underline{221} \\ \underline{3 \ \ \ } | \underline{51} \\ \underline{\ \ \ \ } | \underline{1} \\$

$\displaystyle \text{ ii) } \frac{115}{138} = \frac{5 \times 23}{2 \times 3 \times 23} = \frac{5}{6}$

 $\underline{5 \ \ } | \underline{115} \\ \underline{ \ \ \ } | \underline{23} \\$$\underline{5 \ \ } | \underline{115} \\ \underline{ \ \ \ } | \underline{23} \\$ $\underline{2 \ \ \ } | \underline{138} \\ \underline{3 \ \ \ } | \underline{69} \\ \underline{\ \ \ \ } | \underline{23} \\$$\underline{2 \ \ \ } | \underline{138} \\ \underline{3 \ \ \ } | \underline{69} \\ \underline{\ \ \ \ } | \underline{23} \\$

$\displaystyle \text{ iii) } \frac{87}{116} = \frac{3 \times 29}{4 \times 29} = \frac{3}{4}$

 $\underline{3 \ \ } | \underline{87} \\ \underline{ \ \ \ } | \underline{29} \\$$\underline{3 \ \ } | \underline{87} \\ \underline{ \ \ \ } | \underline{29} \\$ $\underline{4 \ \ } | \underline{116} \\ \underline{ \ \ \ } | \underline{29} \\$$\underline{4 \ \ } | \underline{116} \\ \underline{ \ \ \ } | \underline{29} \\$

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Question 6: Compare the given fractions

$\displaystyle \text{ i) } \frac{13 }{ 14} \text{ and } \frac{20 }{ 21}$    $\displaystyle \text{ ii) } \frac{9 }{ 16} \text{ and } \frac{21 }{ 40}$   $\displaystyle \text{ iii) } \frac{16 }{ 19} \text{ and } \frac{20 }{ 23}$    $\displaystyle \text{ iv) } \frac{10 }{ 11} \text{ and } \frac{18 }{ 19}$

$\displaystyle \text{ i) } \frac{13 }{ 14} \text{ and } \frac{20 }{ 21}$

$\displaystyle \text{ L.C.M of } 14 \text{ and } 21 = 42$

Therefore the fractions can be written as

$\displaystyle \frac{13 }{ 14} = \frac{39 }{ 42} \text{ and } \frac{20 }{ 21} = \frac{40 }{ 42}$

$\displaystyle \text{ Hence, } \frac{13 }{ 14} < \frac{20 }{ 21}$

$\displaystyle \text{ ii) } \frac{9 }{ 16} \text{ and } \frac{21 }{ 40}$

$\displaystyle \text{ L.C.M of } 16 \text{ and } 40 = 80$

Therefore the fractions can be written as

$\displaystyle \frac{9 }{ 16} = \frac{45 }{ 80} \text{ and } \frac{21 }{ 40} = \frac{42 }{ 80}$

$\displaystyle \text{ Hence, } \frac{9 }{ 16} > \frac{21 }{ 40}$

$\displaystyle \text{ iii) } \frac{16 }{ 19} \text{ and } \frac{20 }{ 23}$

$\displaystyle \text{ L.C.M of } 19 \text{ and } 23 = 437$

Therefore the fractions can be written as

$\displaystyle \frac{16 }{ 19} = \frac{368 }{ 437} \text{ and } \frac{20 }{ 23} = \frac{380 }{ 437}$

$\displaystyle \text{ Hence, } \frac{16 }{ 19} < \frac{20 }{ 23}$

$\displaystyle \text{ iv) } \frac{10 }{ 11} \text{ and } \frac{18 }{ 19}$

$\displaystyle \text{ L.C.M of } 11 \text{ and } 19 = 209$

Therefore the fractions can be written as

$\displaystyle \frac{10 }{ 11} = \frac{190 }{ 209} \text{ and } \frac{18 }{ 19} = \frac{198 }{ 209}$

$\displaystyle \text{ Hence, } \frac{10 }{ 11} < \frac{18 }{ 19}$

$\displaystyle \\$

Question 7: Write the following fractions in ascending order of magnitude by making denominators equal.

$\displaystyle \text{ i) } \frac{5 }{ 6} , \frac{7 }{ 9} , \frac{11 }{ 12} , \frac{13 }{ 18}$     $\displaystyle \text{ ii) } \frac{10 }{ 21} , \frac{13 }{ 28} , \frac{26 }{ 35} , \frac{29 }{ 42}$

$\displaystyle \text{ i) } \frac{5 }{ 6} , \frac{7 }{ 9} , \frac{11 }{ 12} , \frac{13 }{ 18}$

$\displaystyle \text{ L.C.M of } 6, 9, 12, 18 = 36$

Hence the fractions can be written as

$\displaystyle \frac{5 }{ 6} = \frac{30 }{ 36} , \frac{7 }{ 9} = \frac{28 }{ 36} , \frac{11 }{ 12} = \frac{33 }{ 36} , \frac{13 }{ 18} = \frac{26 }{ 36}$

$\displaystyle \text{ Hence the ascending order is } \frac{13 }{ 18} < \frac{7 }{ 9} < \frac{5 }{ 6} < \frac{11 }{ 12}$

$\displaystyle \text{ ii) } \frac{10 }{ 21} , \frac{13 }{ 28} , \frac{26 }{ 35} , \frac{29 }{ 42}$

$\displaystyle \text{ L.C.M of } 21, 28, 35, 42 = 420$

Hence the fractions can be written as

$\displaystyle \frac{10 }{ 21} = \frac{200 }{ 420} , \frac{13 }{ 28} = \frac{195 }{ 420} , \frac{26 }{ 35} = \frac{312 }{ 420} , \frac{29 }{ 42} = \frac{290 }{ 420}$

$\displaystyle \text{ Hence the ascending order is } \frac{13 }{ 28} < \frac{10 }{ 21} < \frac{29 }{ 42} < \frac{26 }{ 35}$

$\displaystyle \\$

Question 8: Arrange the following fractions in descending order of Magnitude by making the denominators same.

$\displaystyle \text{ i) } \frac{7 }{ 10} , \frac{13 }{ 15} , \frac{17 }{ 20} , \frac{21 }{ 25} , \frac{31 }{ 50}$     $\displaystyle \text{ ii) } \frac{5 }{ 7} , \frac{9 }{ 14} , \frac{16 }{ 21} , \frac{23 }{ 28} , \frac{29 }{ 42}$

$\displaystyle \text{ i) } \frac{7 }{ 10} , \frac{13 }{ 15} , \frac{17 }{ 20} , \frac{21 }{ 25} , \frac{31 }{ 50}$

$\displaystyle \text{ L.C.M of } 10, 15, 20, 25 \text{ and } 50 = 300$

Hence the fractions can be written as:

$\displaystyle \frac{7 }{ 10} = \frac{210 }{ 300} , \frac{13 }{ 15} = \frac{260 }{ 300} , \frac{17 }{ 20} = \frac{255 }{ 300} , \frac{21 }{ 25} = \frac{252 }{ 300} , \frac{31 }{ 50} = \frac{186 }{ 300 }$

$\displaystyle \text{ Hence the descending order is } \frac{13 }{ 15} > \frac{17 }{ 20} > \frac{21 }{ 25} > \frac{7 }{ 10} > \frac{31 }{ 50}$

$\displaystyle \text{ ii) } \frac{5 }{ 7} , \frac{9 }{ 14} , \frac{16 }{ 21} , \frac{23 }{ 28} , \frac{29 }{ 42}$

$\displaystyle \text{ L.C.M of } 7, 14, 21, 28 \text{ and } 42 = 84$

Hence the fractions can be written as:

$\displaystyle \frac{5 }{ 7} = \frac{60 }{ 84} , \frac{9 }{ 14} = \frac{54 }{ 84} , \frac{16 }{ 21} = \frac{64 }{ 84} , \frac{23 }{ 28} = \frac{69 }{ 84} , \frac{29 }{ 42} = \frac{58 }{ 84 }$

$\displaystyle \text{ Hence the descending order is } \frac{23 }{ 28} > \frac{16 }{ 21} > \frac{5 }{ 7} > \frac{29 }{ 42} > \frac{9 }{ 14}$

$\displaystyle \\$

Question 9: Write the following fractions in descending order by making their numerators the same.

$\displaystyle \text{ i) } \frac{9 }{ 13} , \frac{18 }{ 25} , \frac{27 }{ 40} , \frac{36 }{ 47}$       $\displaystyle \text{ ii) } \frac{10 }{ 17} , \frac{20 }{ 37} , \frac{30 }{ 47} , \frac{40 }{ 53} , \frac{50 }{ 61}$

$\displaystyle \text{ i) } \frac{9 }{ 13} , \frac{18 }{ 25} , \frac{27 }{ 40} , \frac{36 }{ 47}$

$\displaystyle \text{ L.C.M of } 9, 18, 27 \text{ and } 36 = 108$

Hence the fractions can be written as:

$\displaystyle \frac{9 }{ 13} = \frac{108 }{ 156} , \frac{18 }{ 25} = \frac{108 }{ 150} , \frac{27 }{ 40} = \frac{108 }{ 160} , \frac{36 }{ 47} = \frac{108 }{ 141}$

$\displaystyle \text{ Hence the descending order is } \frac{36 }{ 47} > \frac{18 }{ 25} > \frac{9 }{ 13} > \frac{27 }{ 40}$

$\displaystyle \text{ ii) } \frac{10 }{ 17} , \frac{20 }{ 37} , \frac{30 }{ 47} , \frac{40 }{ 53} , \frac{50 }{ 61}$

$\displaystyle \text{ L.C.M of } 10, 20, 30, 40 \text{ and } 50 = 600$

Hence the fractions can be written as:

$\displaystyle \frac{10 }{ 17} = \frac{600 }{ 1020} , \frac{20 }{ 37} = \frac{600 }{ 1110} , \frac{30 }{ 47} = \frac{600 }{ 940} , \frac{40 }{ 53} = \frac{600 }{ 795} , \frac{50 }{ 61} = \frac{600 }{ 793}$

$\displaystyle \text{ Hence the descending order is } \frac{50 }{ 61} > \frac{40 }{ 53} > \frac{30 }{ 47} > \frac{10 }{ 17 } > \frac{20 }{ 37}$

$\displaystyle \\$

Question 10: Insert two fractions between $\displaystyle \frac{4 }{ 7} \text{ and } \frac{2 }{ 3}$

$\displaystyle \text{The fraction between } \frac{4 }{ 7} \text{ and } \frac{2 }{ 3} = \frac{ (4+2) }{ (7+3) } = \frac{6 }{ 10} = \frac{3 }{ 5}$

$\displaystyle \text{The fraction between } \frac{3 }{ 5} \text{ and } \frac{2 }{ 3} = \frac{ (3+2) }{ (5+3) } = \frac{5 }{ 8}$

$\displaystyle \text{ Hence the two fractions are } \frac{3 }{ 5} \text{ and } \frac{5 }{ 8}$

$\displaystyle \\$

Question 11: Insert two fractions between $\displaystyle \frac{8 }{ 11} \text{ and } \frac{11 }{ 6}$

$\displaystyle \text{The fraction between } \frac{8 }{ 11} \text{ and } \frac{11 }{ 6} = \frac{ (8+11) }{ (11+6) } = \frac{19 }{ 17}$
$\displaystyle \text{The fraction between } \frac{19 }{ 17} \text{ and } \frac{11 }{ 6} = \frac{ (19+11) }{ (17+6) } = \frac{30 }{ 23}$
$\displaystyle \text{ Hence the two fractions are } \frac{19 }{ 17} \text{ and } \frac{30 }{ 23}$