Question 1: Convert each of the following fractions into a decimal:

$\displaystyle \text{ i) } \frac{3}{8}$     $\displaystyle \text{ ii) } \frac{15}{16}$    $\displaystyle \text{ iii) } \frac{101}{25}$    $\displaystyle \text{ iv) } \frac{9}{16}$    $\displaystyle \text{ iv) } \frac{13}{40}$    $\displaystyle \text{ vi) } \frac{45}{32}$

$\displaystyle \text{ i) } \frac{3}{8} = 0.375$          $\displaystyle \text{ ii) } \frac{15}{16} = 0.9375$

$\displaystyle \text{ iii) } \frac{101}{25} = 4.04$         $\displaystyle \text{ iv) } \frac{9}{16} = 0.5625$

$\displaystyle \text{ v) } \frac{13}{40} = 0.325$         $\displaystyle \text{ vi) } \frac{45}{32} =1.40625$

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Question 2: Round off:

$\displaystyle \text{ i) } 0.00105 \text{ , correct to } 4 \text{ decimal places}$

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

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

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

$\displaystyle \text{ i) } 0.00105 \text{ , correct to } 4 \text{ decimal places } = 0.0011$

$\displaystyle \text{ ii) } 10.749 \text{ , correct to } 2 \text{ decimal places } = 10.75$

$\displaystyle \text{ iii) } 47.4535 \text{ , correct to } 3 \text{ decimal places } = 47.454$

$\displaystyle \text{ iv) } 182.06451 \text{ , correct to } 2 \text{ decimal places } = 182.06$

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Question 3: Express each of the following as a decimal correct to 3 decimal places :

$\displaystyle \text{ i) } \frac{3}{7}$          $\displaystyle \text{ ii) } \frac{14}{19}$          $\displaystyle \text{ iii) } \frac{19}{21}$          $\displaystyle \text{ iv) } \frac{20}{23}$

$\displaystyle \text{ i) } \frac{3}{7} = 0.429$    $\displaystyle \text{ ii) } \frac{14}{19} = 0.737$    $\displaystyle \text{ iii) } \frac{19}{21} = 0.905$    $\displaystyle \text{ iv) } \frac{20}{23} =0.870$

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Question 4: Express each of the following as a recurring decimal. State in each case, whether it is a pure or a mixed recurring decimal:

$\displaystyle \text{ i) } \frac{1}{6}$    $\displaystyle \text{ ii) } \frac{2}{11}$      $\displaystyle \text{ iii) } \frac{3}{7}$      $\displaystyle \text{ iv) } \frac{8}{18}$

$\displaystyle \text{ i) } \frac{1}{6} =1\dot{6} \text{ This is a mixed recurring decimal. }$

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

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

$\displaystyle \text{ iv) } \frac{8}{18} = \dot{4} \text{ This is a pure recurring decimal. }$

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Question 5: Convert each of the following decimals into a fraction :

(i) $\displaystyle 0.007$           (ii) $\displaystyle 0.0875$          (iii) $\displaystyle 9.005$         (iv) $\displaystyle 4.096$

$\displaystyle \text{ i) } 0.007 = \frac{7}{1000}$

$\displaystyle \text{ ii) } 0.0875 = \frac{875}{10000} = \frac{175}{2000} = \frac{7}{40}$

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

$\displaystyle \text{ iv) } 4.096 = \frac{4096}{1000} = \frac{2048}{500} = \frac{1024}{250} = \frac{512}{125}$

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Question 6: Express each of the following decimals as a rational number:

(i) $\displaystyle 0. \dot{7}$     (ii) $\displaystyle 0.\overline{36}$     (iii) $\displaystyle 4.\overline{26}$     (iv) $\displaystyle 0. 6\dot{3}$     (v) $\displaystyle 0.22\overline{73}$     (vi) $\displaystyle 2.1\overline{36}$

 (i) $0. \dot{7}$$0. \dot{7}$ Let $x=0.7777777 \ldots$$x=0.7777777 \ldots$ $\Rightarrow 10x=7.777777 \ldots$$\Rightarrow 10x=7.777777 \ldots$ $\displaystyle \Rightarrow 9x = 7 \ or \ x = \frac{7}{9}$$\displaystyle \Rightarrow 9x = 7 \ or \ x = \frac{7}{9}$ (ii) $0.\overline{36}$$0.\overline{36}$ Let $x=0.363636 \ldots$$x=0.363636 \ldots$ $\Rightarrow 100x=36.363636 \ldots$$\Rightarrow 100x=36.363636 \ldots$ $\Rightarrow 99x=36$$\Rightarrow 99x=36$ $\displaystyle \ or \ x= \frac{4}{11}$$\displaystyle \ or \ x= \frac{4}{11}$ (iii) $4.\overline{26}$$4.\overline{26}$ Let $x=4.26262626 \ldots$$x=4.26262626 \ldots$ $\Rightarrow 100x=426.262626 \ldots$$\Rightarrow 100x=426.262626 \ldots$ $\Rightarrow 99x=422$$\Rightarrow 99x=422$ $\displaystyle \ or \ x= \frac{422}{99}$$\displaystyle \ or \ x= \frac{422}{99}$ (iv) $0. 6\dot{3}$$0. 6\dot{3}$ Let $x=0.63333333 \ldots$$x=0.63333333 \ldots$ $\Rightarrow 10x=6.3333333 \ldots$$\Rightarrow 10x=6.3333333 \ldots$ $\Rightarrow 100x=63.333333 \ldots$$\Rightarrow 100x=63.333333 \ldots$ $\Rightarrow 90x=57$$\Rightarrow 90x=57$ $\displaystyle \ or \ x= \frac{19}{30}$$\displaystyle \ or \ x= \frac{19}{30}$ (v) $0.22\overline{73}$$0.22\overline{73}$ Let $x=0.227373737373 \ldots$$x=0.227373737373 \ldots$ $\Rightarrow 100x=22.73737373737 \ldots$$\Rightarrow 100x=22.73737373737 \ldots$ $\Rightarrow 10000x=2273.7373737373 \ldots$$\Rightarrow 10000x=2273.7373737373 \ldots$ $\Rightarrow 9900x=2251$$\Rightarrow 9900x=2251$ $\displaystyle \ or \ x= \frac{2251}{9900}$$\displaystyle \ or \ x= \frac{2251}{9900}$ (vi) $2.1\overline{36}$$2.1\overline{36}$ Let $x=2.13636363636 \ldots$$x=2.13636363636 \ldots$ $\Rightarrow 10x=21.3636363636 \ldots$$\Rightarrow 10x=21.3636363636 \ldots$ $\Rightarrow 1000x=2136.36363636 \ldots$$\Rightarrow 1000x=2136.36363636 \ldots$ $\Rightarrow 990 x = 2115$$\Rightarrow 990 x = 2115$ $\displaystyle \ or \ x = \frac{2115}{990} = \frac{235}{110}$$\displaystyle \ or \ x = \frac{2115}{990} = \frac{235}{110}$

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Question 7: Write the following fractions in descending order by converting them into decimals:

(i) $\frac{8}{25}$ $,$ $\frac{4}{15}$ $,$ $\frac{9}{20}$ $,$ $\frac{3}{8}$ $,$ $\frac{11}{16}$

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

(i) $\frac{8}{25}$ $,$ $\frac{4}{15}$ $,$ $\frac{9}{20}$ $,$ $\frac{3}{8}$ $,$ $\frac{11}{16}$

$\Rightarrow 0.3200, 0.2667, 0.4500, 0.3750, 0.6875$

Now arranging in descending order: $\frac{11}{16}$ $,$ $\frac{9}{20}$ $,$ $\frac{3}{8}$ $,$ $\frac{8}{25}$ $,$ $\frac{4}{15}$

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

$\Rightarrow 0.7333, 0.7059, 2.0714, 0.7083, 0.7368$

Now arranging in descending order: $\frac{29}{14}$ $,$ $\frac{14}{19}$ $,$ $\frac{11}{15}$ $,$ $\frac{17}{24}, \frac{12}{19}$

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Question 8: Find the H. C.F. and L.C.M. of:

(i) $0.63, 1.05, 2.1$     (ii) $1.08, 0.36, 0.9$    (iii) $0.3, 0.03, 3$    (iv) $1.75, 5.6, 7$

(i) $0.63, 1.05, 2.1$

Convert the numbers into like decimals i.e. $0.63, 1.05, 2.10$

First find HCF and LCM of $63, 105$, and $210$

HCF $= 21$

LCM $= 630$

Therefore HCF and LCM of $0.63, 1.05$, and $2.1$ is $0.21$ and $6.30$ respectively.

(ii) $1.08, 0.36, 0.9$

Convert the numbers into like decimals. i.e. $1.08, 0.36, 0.90$

First find HCF and LCM of $108, 36$, and $90$

HCF $= 18$

LCM $= 540$

Therefore the HCF and LCM of $1.08, 0.36$ and $0.9$ is $0.18$ and $5.40$ respectively.

(iii) $0.3, 0.03, 3$

Convert the numbers into like decimals $0.30, 0.03, 3.00$

Find HCF and LCM of $30, 3$ and $300$

HCF $= 3$

LCM $= 300$

Therefore the HCF and LCM of $0.3, 0.03$ and $3$ is $0.03$ and $3.00$ respectively.

(iv) $1.75, 5.6, 7$

Convert the numbers into like decimals i.e. $1.75, 5.60, 7.00$

Then find the HCF and LCM of $175, 560$ and $700$

HCF $= 35$

LCM $= 2800$

Therefore the HCF and LCM of $1.75, 5.6$ and $7$ is $0.35$ and $28.00$