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}   

Answer:

\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} 

Answer:

i) \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}

ii) \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}

iii) \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}

iv) \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}

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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}   

Answer:

\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}   

Answer:

\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}    

Answer:

\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{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{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{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} 

Answer:

\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} 

Answer:

\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} 

Answer:

\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} 

Answer:

\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} 

Answer:

\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} 

Answer:

\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}