Solved examples on rational numbers

Here we are giving some practice  questions based  on  rational numbers.

1. Are the following statements are true or false? Give reason  for your answers.

(i)  every whole number is a natural numbers.

(ii) every integer is a rational number .

(iii)There are infinitely many rational numbers between any two given rational numbers.

(iv) 0 is a rational  number.

(V)Every rational number is a whole number.

ANS. (i) False , beacuse ‘0’ is a whole  number not a natural number.

(ii)True,  because every integer m can be expressed in the form \frac{m}{1}, so it is a rational number.

(iii) True

(iv) True , since we can write  0=\frac{0}{1}

(v)false

2. Find six rational numbers between 3 and 4. 

Sol.  To find a rational  number between  r and  s we can add r and s and divide the sum by 2 that is  \frac{r+s}{2}  lies between  r and  s .

Now let  r =3  and  s =4 , then \frac{r+s}{2}  = \frac{7}{2} is a number lying between  3 and 4 .

Next let  r =3  and  s =\frac{7}{2}, then \frac{3+\frac{7}{2}}{2}= \frac{13}{4}  is also a number lying between  3 and 4.

Rational no. between \frac{7}{2} \; and \; 4 is  \frac{\frac{7}{2}+4}{2}= \frac{15}{4}  .

Proceeding in the same manner  the six  rational  numbers  between 3 and 4 are \frac{7}{2},  \frac{13}{4} , \frac{15}{4},\frac{27}{8}, \frac{28}{8}, \frac{29}{8}.

 

3. Express the decimal    32.12\bar{35}    in the form \frac{p}{q}  where p  and q are integers and q\neq 0.

Sol. let    x=32.12353535.......  Since two digits are repeating, we multiply by x by 100 to get

100x= 3212.353535..........

So,      100x=3180.23+32.123535........... \: =3180.23+x

Therefore           100x-x =3180.23

i.e. 99x= \frac{318023}{100}  , which gives  x=\frac{318023}{9900}

Hence , 32.12\bar{35} =\frac{318023}{9900}.

 

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