\displaystyle \textbf{Question 1: }\text{Express each of the following as an improper fraction:}
\displaystyle \text{(i) }9\frac{14}{15}\qquad\text{(ii) }17\frac{5}{6}\qquad\text{(iii) }13\frac{11}{26}\qquad\text{(iv) }3\frac{41}{51}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }9\frac{14}{15}=\frac{9\times15+14}{15}=\frac{149}{15}.

\displaystyle \text{(ii) }17\frac{5}{6}=\frac{17\times6+5}{6}=\frac{107}{6}.

\displaystyle \text{(iii) }13\frac{11}{26}=\frac{13\times26+11}{26}=\frac{349}{26}.

\displaystyle \text{(iv) }3\frac{41}{51}=\frac{3\times51+41}{51}=\frac{194}{51}.

\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Express each of the following as a mixed fraction:}
\displaystyle \text{(i) }\frac{135}{26}\qquad\text{(ii) }\frac{148}{35}\qquad\text{(iii) }\frac{620}{17}\qquad\text{(iv) }\frac{1075}{96}
\displaystyle \text{Answer:}
\displaystyle \text{(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}

\displaystyle \text{(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}

\displaystyle \text{(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}

\displaystyle \text{(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}

\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Write five fractions equivalent to each of the following fractions:}
\displaystyle \text{(i) }\frac{4}{7}\qquad\text{(ii) }\frac{128}{192}
\displaystyle \text{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 \\

\displaystyle \textbf{Question 4: }\text{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{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 \frac{3}{4}=\frac{18}{24},\qquad\frac{5}{8}=\frac{15}{24},\qquad\frac{7}{12}=\frac{14}{24},\qquad\frac{13}{24}=\frac{13}{24}.
\displaystyle \therefore \text{The required like fractions are }\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 \frac{1}{3}=\frac{10}{30},\qquad\frac{4}{5}=\frac{24}{30},\qquad\frac{7}{10}=\frac{21}{30},\qquad\frac{11}{15}=\frac{22}{30}.
\displaystyle \therefore \text{The required like fractions are }\frac{10}{30},\ \frac{24}{30},\ \frac{21}{30},\ \frac{22}{30}.

\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Reduce each of the following to the simplest form:}
\displaystyle \text{(i) }\frac{153}{221}\qquad\text{(ii) }\frac{115}{138}\qquad\text{(iii) }\frac{87}{116}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{153}{221}=\frac{3\times3\times17}{17\times13}=\frac{9}{13}.
\underline{3\ \ \ }|\underline{153} \\\ \underline{3\ \ \ }|\underline{51} \\\ \underline{17\ }|\underline{17} \\\ \underline{\ \ \ \ }|\underline{1}
\underline{17\ }|\underline{221} \\\ \underline{13\ }|\underline{13} \\\ \underline{\ \ \ \ }|\underline{1}

\displaystyle \text{(ii) }\frac{115}{138}=\frac{5\times23}{2\times3\times23}=\frac{5}{6}.
\underline{5\ \ }|\underline{115} \\\ \underline{23\ }|\underline{23} \\\ \underline{\ \ \ }|\underline{1}
\underline{2\ \ }|\underline{138} \\\ \underline{3\ \ }|\underline{69} \\\ \underline{23\ }|\underline{23} \\\ \underline{\ \ \ }|\underline{1}

\displaystyle \text{(iii) }\frac{87}{116}=\frac{3\times29}{2\times2\times29}=\frac{3}{4}.
\underline{3\ \ }|\underline{87} \\\ \underline{29\ }|\underline{29} \\\ \underline{\ \ \ }|\underline{1}
\underline{2\ \ }|\underline{116} \\\ \underline{2\ \ }|\underline{58} \\\ \underline{29\ }|\underline{29} \\\ \underline{\ \ \ }|\underline{1}

\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Compare the given fractions:}
\displaystyle \text{(i) }\frac{13}{14}\text{ and }\frac{20}{21}\qquad\text{(ii) }\frac{9}{16}\text{ and }\frac{21}{40}
\displaystyle \text{(iii) }\frac{16}{19}\text{ and }\frac{20}{23}\qquad\text{(iv) }\frac{10}{11}\text{ and }\frac{18}{19}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{13}{14}\text{ and }\frac{20}{21}
\displaystyle \text{L.C.M. of }14\text{ and }21=42.
\displaystyle \frac{13}{14}=\frac{39}{42}\qquad\text{and}\qquad\frac{20}{21}=\frac{40}{42}.
\displaystyle \text{Since }39<40,\text{ we have }\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.
\displaystyle \frac{9}{16}=\frac{45}{80}\qquad\text{and}\qquad\frac{21}{40}=\frac{42}{80}.
\displaystyle \text{Since }45>42,\text{ we have }\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.
\displaystyle \frac{16}{19}=\frac{368}{437}\qquad\text{and}\qquad\frac{20}{23}=\frac{380}{437}.
\displaystyle \text{Since }368<380,\text{ we have }\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.
\displaystyle \frac{10}{11}=\frac{190}{209}\qquad\text{and}\qquad\frac{18}{19}=\frac{198}{209}.
\displaystyle \text{Since }190<198,\text{ we have }\frac{10}{11}<\frac{18}{19}.

\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Write the following fractions in ascending order of magnitude by making the 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{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.
\displaystyle \text{Hence, the fractions can be written as:}
\displaystyle \frac{5}{6}=\frac{30}{36},\qquad\frac{7}{9}=\frac{28}{36},\qquad\frac{11}{12}=\frac{33}{36},\qquad\frac{13}{18}=\frac{26}{36}.
\displaystyle \text{Since }26<28<30<33,
\displaystyle \therefore \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.
\displaystyle \text{Hence, the fractions can be written as:}
\displaystyle \frac{10}{21}=\frac{200}{420},\qquad\frac{13}{28}=\frac{195}{420},\qquad\frac{26}{35}=\frac{312}{420},\qquad\frac{29}{42}=\frac{290}{420}.
\displaystyle \text{Since }195<200<290<312,
\displaystyle \therefore \frac{13}{28}<\frac{10}{21}<\frac{29}{42}<\frac{26}{35}.

\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Arrange the following fractions in descending order of magnitude by making the denominators equal.}
\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{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,\ 50=300.
\displaystyle \text{Hence, the fractions can be written as:}
\displaystyle \frac{7}{10}=\frac{210}{300},\qquad\frac{13}{15}=\frac{260}{300},\qquad\frac{17}{20}=\frac{255}{300},
\displaystyle \frac{21}{25}=\frac{252}{300},\qquad\frac{31}{50}=\frac{186}{300}.
\displaystyle \text{Since }260>255>252>210>186,
\displaystyle \therefore \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,\ 42=84.
\displaystyle \text{Hence, the fractions can be written as:}
\displaystyle \frac{5}{7}=\frac{60}{84},\qquad\frac{9}{14}=\frac{54}{84},\qquad\frac{16}{21}=\frac{64}{84},
\displaystyle \frac{23}{28}=\frac{69}{84},\qquad\frac{29}{42}=\frac{58}{84}.
\displaystyle \text{Since }69>64>60>58>54,
\displaystyle \therefore \frac{23}{28}>\frac{16}{21}>\frac{5}{7}>\frac{29}{42}>\frac{9}{14}.

\displaystyle \\

\displaystyle \textbf{Question 9: }\text{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{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,\ 36=108.
\displaystyle \text{Hence, the fractions can be written as:}
\displaystyle \frac{9}{13}=\frac{108}{156},\qquad\frac{18}{25}=\frac{108}{150},\qquad\frac{27}{40}=\frac{108}{160},\qquad\frac{36}{47}=\frac{108}{141}.
\displaystyle \text{For fractions having the same numerator, the fraction with the smaller denominator is larger.}
\displaystyle \text{Since }141<150<156<160,
\displaystyle \therefore \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,\ 50=600.
\displaystyle \text{Hence, the fractions can be written as:}
\displaystyle \frac{10}{17}=\frac{600}{1020},\qquad\frac{20}{37}=\frac{600}{1110},\qquad\frac{30}{47}=\frac{600}{940},
\displaystyle \frac{40}{53}=\frac{600}{795},\qquad\frac{50}{61}=\frac{600}{732}.
\displaystyle \text{For fractions having the same numerator, the fraction with the smaller denominator is larger.}
\displaystyle \text{Since }732<795<940<1020<1110,
\displaystyle \therefore \frac{50}{61}>\frac{40}{53}>\frac{30}{47}>\frac{10}{17}>\frac{20}{37}.

\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Insert two fractions between }\frac{4}{7}\text{ and }\frac{2}{3}.
\displaystyle \text{Answer:}
\displaystyle \text{A fraction between }\frac{4}{7}\text{ and }\frac{2}{3}\text{ is }\frac{4+2}{7+3}=\frac{6}{10}=\frac{3}{5}.
\displaystyle \text{A fraction between }\frac{3}{5}\text{ and }\frac{2}{3}\text{ is }\frac{3+2}{5+3}=\frac{5}{8}.
\displaystyle \therefore \text{The two fractions are }\frac{3}{5}\text{ and }\frac{5}{8}.
\displaystyle \text{Indeed, }\frac{4}{7}<\frac{3}{5}<\frac{5}{8}<\frac{2}{3}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Insert two fractions between }\frac{8}{11}\text{ and }\frac{11}{6}.
\displaystyle \text{Answer:}
\displaystyle \text{A fraction between }\frac{8}{11}\text{ and }\frac{11}{6}\text{ is }\frac{8+11}{11+6}=\frac{19}{17}.
\displaystyle \text{A fraction between }\frac{19}{17}\text{ and }\frac{11}{6}\text{ is }\frac{19+11}{17+6}=\frac{30}{23}.
\displaystyle \therefore \text{The two fractions are }\frac{19}{17}\text{ and }\frac{30}{23}.
\displaystyle \text{Indeed, }\frac{8}{11}<\frac{19}{17}<\frac{30}{23}<\frac{11}{6}.
\displaystyle \\

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