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}   

Answer:

\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

\displaystyle  \\

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}

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  \\

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}   

Answer:

\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

\displaystyle  \\

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}   

Answer:

\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. }

\displaystyle  \\

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 

Answer:

\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}   

\displaystyle  \\

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}

Answer:

(i) 0. \dot{7}

Let x=0.7777777 \ldots

\Rightarrow 10x=7.777777 \ldots

\displaystyle  \Rightarrow 9x = 7 \ or \ x =   \frac{7}{9} 

(ii) 0.\overline{36}

Let x=0.363636 \ldots

\Rightarrow 100x=36.363636 \ldots

\Rightarrow 99x=36

\displaystyle  \ or \ x= \frac{4}{11}

(iii) 4.\overline{26}

Let x=4.26262626 \ldots

\Rightarrow 100x=426.262626 \ldots

\Rightarrow 99x=422

\displaystyle \ or \ x=  \frac{422}{99}

(iv) 0. 6\dot{3}

Let x=0.63333333 \ldots

\Rightarrow 10x=6.3333333 \ldots

\Rightarrow 100x=63.333333 \ldots

\Rightarrow 90x=57

\displaystyle \ or \ x=  \frac{19}{30}

(v) 0.22\overline{73}

Let x=0.227373737373 \ldots

\Rightarrow 100x=22.73737373737 \ldots

\Rightarrow 10000x=2273.7373737373 \ldots

\Rightarrow 9900x=2251

\displaystyle \ or \ x=  \frac{2251}{9900}

(vi) 2.1\overline{36}

Let x=2.13636363636 \ldots

\Rightarrow 10x=21.3636363636 \ldots

\Rightarrow 1000x=2136.36363636 \ldots

\Rightarrow 990 x = 2115

\displaystyle \ or \ x =  \frac{2115}{990}  =  \frac{235}{110}

\displaystyle  \\

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}

Answer:

(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}

\displaystyle  \\

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

Answer:

(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