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) $3$ $\frac{41}{51}$

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) $\frac{1075}{96}$

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

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

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

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) $\frac{87}{116}$

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} \\$

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} \\$

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

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 $\frac{18 }{ 19}$

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

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

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

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 $\frac{30 }{ 23}$