\displaystyle \textbf{Question 1: }\text{Convert each of the following fractions into a decimal:}
\displaystyle \text{(i) }\frac{3}{8}\qquad\text{(ii) }\frac{15}{16}\qquad\text{(iii) }\frac{101}{25}

\displaystyle \text{(iv) }\frac{9}{16}\qquad\text{(v) }\frac{13}{40}\qquad\text{(vi) }\frac{45}{32}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{3}{8}=3\div8=0.375.

\displaystyle \text{(ii) }\frac{15}{16}=15\div16=0.9375.

\displaystyle \text{(iii) }\frac{101}{25}=101\div25=4.04.

\displaystyle \text{(iv) }\frac{9}{16}=9\div16=0.5625.

\displaystyle \text{(v) }\frac{13}{40}=13\div40=0.325.

\displaystyle \text{(vi) }\frac{45}{32}=45\div32=1.40625.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Round off:}
\displaystyle \text{(i) }0.00105,\text{ correct to }4\text{ decimal places}

\displaystyle \text{(ii) }10.749,\text{ correct to }2\text{ decimal places}

\displaystyle \text{(iii) }47.4535,\text{ correct to }3\text{ decimal places}

\displaystyle \text{(iv) }182.06451,\text{ correct to }2\text{ decimal places}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }0.00105\text{ correct to }4\text{ decimal places}=0.0011.

\displaystyle \text{(ii) }10.749\text{ correct to }2\text{ decimal places}=10.75.

\displaystyle \text{(iii) }47.4535\text{ correct to }3\text{ decimal places}=47.454.

\displaystyle \text{(iv) }182.06451\text{ correct to }2\text{ decimal places}=182.06.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Express each of the following as a decimal correct to }
\displaystyle 3\text{ decimal places:}
\displaystyle \text{(i) }\frac{3}{7}\qquad\text{(ii) }\frac{14}{19}\qquad\text{(iii) }\frac{19}{21}

\displaystyle \text{(iv) }\frac{20}{23}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{3}{7}=0.428571\ldots=0.429\text{ (correct to 3 decimal places).}

\displaystyle \text{(ii) }\frac{14}{19}=0.736842\ldots=0.737\text{ (correct to 3 decimal places).}

\displaystyle \text{(iii) }\frac{19}{21}=0.904761\ldots=0.905\text{ (correct to 3 decimal places).}

\displaystyle \text{(iv) }\frac{20}{23}=0.869565\ldots=0.870\text{ (correct to 3 decimal places).}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Express each of the following as a recurring decimal.}
\displaystyle \text{State in each case whether it is a pure or a mixed recurring decimal:}
\displaystyle \text{(i) }\frac{1}{6}\qquad\text{(ii) }\frac{2}{11}\qquad\text{(iii) }\frac{3}{7}

\displaystyle \text{(iv) }\frac{8}{18}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{1}{6}=0.1\overline{6}.
\displaystyle \text{This is a mixed recurring decimal.}

\displaystyle \text{(ii) }\frac{2}{11}=0.\overline{18}.
\displaystyle \text{This is a pure recurring decimal.}

\displaystyle \text{(iii) }\frac{3}{7}=0.\overline{428571}.
\displaystyle \text{This is a pure recurring decimal.}

\displaystyle \text{(iv) }\frac{8}{18}=\frac{4}{9}=0.\overline{4}.
\displaystyle \text{This is a pure recurring decimal.}
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Convert each of the following decimals into a fraction:}
\displaystyle \text{(i) }0.007\qquad\text{(ii) }0.0875\qquad\text{(iii) }9.005

\displaystyle \text{(iv) }4.096

\displaystyle \text{Answer:}
\displaystyle \text{(i) }0.007=\frac{7}{1000}.

\displaystyle \text{(ii) }0.0875=\frac{875}{10000}=\frac{7}{80}.

\displaystyle \text{(iii) }9.005=\frac{9005}{1000}=\frac{1801}{200}.

\displaystyle \text{(iv) }4.096=\frac{4096}{1000}=\frac{512}{125}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Express each of the following decimals as a rational number:}
\displaystyle \text{(i) }0.\overline{7}\qquad\text{(ii) }0.\overline{36}\qquad\text{(iii) }4.\overline{26}

\displaystyle \text{(iv) }0.6\overline{3}\qquad\text{(v) }0.22\overline{73}\qquad\text{(vi) }2.1\overline{36}

\displaystyle \text{Answer:}
\displaystyle \text{(i) Let }x=0.\overline{7}=0.777777\ldots
\displaystyle \therefore 10x=7.777777\ldots
\displaystyle 10x-x=7.777777\ldots-0.777777\ldots
\displaystyle 9x=7
\displaystyle \therefore x=\frac{7}{9}.

\displaystyle \text{(ii) Let }x=0.\overline{36}=0.363636\ldots
\displaystyle \therefore 100x=36.363636\ldots
\displaystyle 100x-x=36.363636\ldots-0.363636\ldots
\displaystyle 99x=36
\displaystyle \therefore x=\frac{36}{99}=\frac{4}{11}.

\displaystyle \text{(iii) Let }x=4.\overline{26}=4.262626\ldots
\displaystyle \therefore 100x=426.262626\ldots
\displaystyle 100x-x=426.262626\ldots-4.262626\ldots
\displaystyle 99x=422
\displaystyle \therefore x=\frac{422}{99}.

\displaystyle \text{(iv) Let }x=0.6\overline{3}=0.633333\ldots
\displaystyle \therefore 10x=6.333333\ldots
\displaystyle \therefore 100x=63.333333\ldots
\displaystyle 100x-10x=63.333333\ldots-6.333333\ldots
\displaystyle 90x=57
\displaystyle \therefore x=\frac{57}{90}=\frac{19}{30}.

\displaystyle \text{(v) Let }x=0.22\overline{73}=0.22737373\ldots
\displaystyle \therefore 100x=22.737373\ldots
\displaystyle \therefore 10000x=2273.737373\ldots
\displaystyle 10000x-100x=2273.737373\ldots-22.737373\ldots
\displaystyle 9900x=2251
\displaystyle \therefore x=\frac{2251}{9900}.

\displaystyle \text{(vi) Let }x=2.1\overline{36}=2.1363636\ldots
\displaystyle \therefore 10x=21.363636\ldots
\displaystyle \therefore 1000x=2136.363636\ldots
\displaystyle 1000x-10x=2136.363636\ldots-21.363636\ldots
\displaystyle 990x=2115
\displaystyle \therefore x=\frac{2115}{990}=\frac{47}{22}.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Write the following fractions in descending order}
\displaystyle \text{by converting them into decimals:}
\displaystyle \text{(i) }\frac{8}{25},\ \frac{4}{15},\ \frac{9}{20},\ \frac{3}{8},\ \frac{11}{16}

\displaystyle \text{(ii) }\frac{11}{15},\ \frac{12}{17},\ \frac{29}{14},\ \frac{17}{24},\ \frac{14}{19}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{8}{25}=0.3200,\quad\frac{4}{15}=0.2667,\quad\frac{9}{20}=0.4500
\displaystyle \frac{3}{8}=0.3750,\qquad\frac{11}{16}=0.6875.
\displaystyle \text{Since }0.6875>0.4500>0.3750>0.3200>0.2667,
\displaystyle \therefore \frac{11}{16}>\frac{9}{20}>\frac{3}{8}>\frac{8}{25}>\frac{4}{15}.

\displaystyle \text{(ii) }\frac{11}{15}=0.7333,\quad\frac{12}{17}=0.7059,\quad\frac{29}{14}=2.0714
\displaystyle \frac{17}{24}=0.7083,\qquad\frac{14}{19}=0.7368.
\displaystyle \text{Since }2.0714>0.7368>0.7333>0.7083>0.7059,
\displaystyle \therefore \frac{29}{14}>\frac{14}{19}>\frac{11}{15}>\frac{17}{24}>\frac{12}{17}.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Find the H.C.F. and L.C.M. of:}
\displaystyle \text{(i) }0.63,\ 1.05,\ 2.1\qquad\text{(ii) }1.08,\ 0.36,\ 0.9

\displaystyle \text{(iii) }0.3,\ 0.03,\ 3\qquad\text{(iv) }1.75,\ 5.6,\ 7

\displaystyle \text{Answer:}
\displaystyle \text{(i) Converting the numbers into like decimals:}
\displaystyle 0.63,\ 1.05,\ 2.10.
\displaystyle \text{Removing the decimal points, we get }63,\ 105,\ 210.
\displaystyle \text{H.C.F. of }63,\ 105,\ 210=21.
\displaystyle \text{L.C.M. of }63,\ 105,\ 210=630.
\displaystyle \therefore \text{H.C.F.}=0.21,\qquad\text{L.C.M.}=6.30.

\displaystyle \text{(ii) Converting the numbers into like decimals:}
\displaystyle 1.08,\ 0.36,\ 0.90.
\displaystyle \text{Removing the decimal points, we get }108,\ 36,\ 90.
\displaystyle \text{H.C.F. of }108,\ 36,\ 90=18.
\displaystyle \text{L.C.M. of }108,\ 36,\ 90=540.
\displaystyle \therefore \text{H.C.F.}=0.18,\qquad\text{L.C.M.}=5.40.

\displaystyle \text{(iii) Converting the numbers into like decimals:}
\displaystyle 0.30,\ 0.03,\ 3.00.
\displaystyle \text{Removing the decimal points, we get }30,\ 3,\ 300.
\displaystyle \text{H.C.F. of }30,\ 3,\ 300=3.
\displaystyle \text{L.C.M. of }30,\ 3,\ 300=300.
\displaystyle \therefore \text{H.C.F.}=0.03,\qquad\text{L.C.M.}=3.00.

\displaystyle \text{(iv) Converting the numbers into like decimals:}
\displaystyle 1.75,\ 5.60,\ 7.00.
\displaystyle \text{Removing the decimal points, we get }175,\ 560,\ 700.
\displaystyle \text{H.C.F. of }175,\ 560,\ 700=35.
\displaystyle \text{L.C.M. of }175,\ 560,\ 700=2800.
\displaystyle \therefore \text{H.C.F.}=0.35,\qquad\text{L.C.M.}=28.00.
\displaystyle \\

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