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

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

Answer:

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

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

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

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

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Question 2: 

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

Answer:

i) \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) \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) \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) \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:

i) \frac{4}{7}     ii) \frac{128 }{ 192}

Answer:

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

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

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Question 4: Convert the unlike fractions into like fractions:

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

Answer:

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

L.C.M of 4, 8, 12, 24 = 24 . Therefore the fractions can be written as following like fractions \frac{18 }{ 24} , \frac{15 }{ 24} , \frac{14 }{ 24} , \frac{13 }{ 24}

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

L.C.M of 3, 5, 10, 15 = 30 . Therefore the fractions can be written as following like fractions \frac{10 }{ 30} , \frac{24 }{ 30} , \frac{21 }{ 30} , \frac{22 }{ 30}

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

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

Answer:

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} \\ 

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} \\ 

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

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

Answer:

i) \frac{13 }{ 14} and \frac{20 }{ 21}

L.C.M of 14 and 21 = 42

Therefore the fractions can be written as

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

Hence, \frac{13 }{ 14} < \frac{20 }{ 21}

ii) \frac{9 }{ 16} and \frac{21 }{ 40}

L.C.M of 16 and 40 = 80

Therefore the fractions can be written as

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

Hence, \frac{9 }{ 16} > \frac{21 }{ 40}

iii) \frac{16 }{ 19} and \frac{20 }{ 23}

L.C.M of 19 and 23 = 437

Therefore the fractions can be written as

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

Hence, \frac{16 }{ 19} < \frac{20 }{ 23}

iv) \frac{10 }{ 11} and \frac{18 }{ 19}

L.C.M of 11 and 19 = 209

Therefore the fractions can be written as

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

Hence, \frac{10 }{ 11} < \frac{18 }{ 19}

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

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

Answer:

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

L.C.M of 6, 9, 12, 18 = 36

Hence the fractions can be written as

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

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

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

L.C.M of 21, 28, 35, 42 = 420

Hence the fractions can be written as

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

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

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

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

Answer:

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

L.C.M of 10, 15, 20, 25 and 50 = 300

Hence the fractions can be written as:

\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 }

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

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

L.C.M of 7, 14, 21, 28 and 42 = 84

Hence the fractions can be written as:

\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 }

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

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

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

Answer:

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

L.C.M of 9, 18, 27 and 36 = 108

Hence the fractions can be written as:

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

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

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

L.C.M of 10, 20, 30, 40 and 50 = 600

Hence the fractions can be written as:

\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}

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

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

Answer: The fraction between \frac{4 }{ 7} and \frac{2 }{ 3} = \frac{ (4+2) }{ (7+3) } = \frac{6 }{ 10} = \frac{3 }{ 5}

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

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

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

Answer: The fraction between \frac{8 }{ 11} and \frac{11 }{ 6} = \frac{ (8+11) }{ (11+6) } = \frac{19 }{ 17}

The fraction between \frac{19 }{ 17} and \frac{11 }{ 6} = \frac{ (19+11) }{ (17+6) } = \frac{30 }{ 23}

Hence the two fractions are \frac{19 }{ 17} and \frac{30 }{ 23}