TOPICS
Exercise - 1.3

Question-1 :-  Write the following in decimal form and say what kind of decimal expansion each has :
(i) 36100 ,  (ii) 111 ,  (iii) 418 ,  (iv) 313 ,  (v) 211 ,  (vi) 329400

Solution :-
 (i)   36/100 = 0.36   (Terminating)
 (ii)  1/11 = 0.09090... = 0.09   (Non-Terminating Repeating)
 (iii) 418 = 33/8 = 4.125   (Terminating)
 (iv)  3/13 = 0.230969230769... = 0.230969   (Non-Terminating Repeating)
 (v)   2/11 = 0.181818... = 0.18   (Non-Terminating Repeating)
 (vi)  329/400 = 0.8225   (Terminating)
    

Question-2 :-  You know that 1/7 = 0.142857 Can you predict what the decimal expansions of 27 , 37 , 47 , 57 , 67 are, without actually doing the long division? If so, how?

Solution :-
(i)   2/7 = 2 x 1/7 = 2 x 0.142857 = 0.285714     
(ii)  3/7 = 3 x 1/7 = 3 x 0.142857 = 0.428571     
(iii) 4/7 = 4 x 1/7 = 2 x 0.142857 = 0.571428     
(iv)  5/7 = 5 x 1/7 = 2 x 0.142857 = 0.714285     
(v)   6/7 = 6 x 1/7 = 2 x 0.142857 = 0.857142     
    

Question-3 :-  Express the following in the form p/q, where p and q are integers and q ≠ 0.
(i) 0.6  ,  (ii) 0.47  ,  (iii) 0.001

Solution :-
(i) Let x = 0.6666... and there is one digit is repeating, so multiply by 10 both sides
    10x = 10 X (0.6666...) = 6.666...
    6.666... = 6 + x, (Since x = 0.666...)
    10x = 6 + x
    10x - x = 6
    9x = 6 , x = 6/9, i.e., x = 2/3

(ii) Let x = 0.4777... and there is two digit is repeating, so multiply by 100 both sides
     10x = 10 X (0.4777...) = 4.777...
     10x = 4.3 + 0.4777...
     10x = 4.3 + x, (Since x = 0.4777...)
     10x - x = 4.3
     9x = 4.3 , x = 4.3/90, i.e., x = 43/90

(iii) Let x = 0.6666... and there is one digit is repeating, so multiply by 10 both sides
    10x = 10 X (0.6666...) = 6.666...
    6.666... = 6 + x, (Since x = 0.666...)
    10x = 6 + x
    10x - x = 6
    9x = 6 , x = 6/9, i.e., x = 2/3
    

Question-4 :-  Express 0.99999 .... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Solution :-
 Let x = 0.9999...  and there is one digit is repeating, so multiply by 10 both sides 
 10x = 1 x 0.9999... = 9.999...  
 10x = 9 + x (Since x = 0.9999...)
 9x = 9
 x = 9/9 
 x = 1
    

Question-5 :-  What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.

Solution :-
 It can be observed that, 1/17 = 0.0588235294117647
 There are 16 digits in the repeating numbers of the decimal expansion of 1/17.   
    

Question-6 :-  Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Solution :-
 Terminating decimal expansion will occur when denominator q of rational number p/q is either of 
 2, 4, 5, 8, 10, and so on...
 9/4 = 2.25
 11/8 = 1.375
 27/5 = 5.4
 It can be observed that terminating decimal may be obtained in the situation where prime factorisation 
 of the denominator of the given fractions has the power of 2 only or 5 only or both.
    

Question-7 :-  Write three numbers whose decimal expansions are non-terminating non-recurring.

Solution :-
 3 numbers whose decimal expansions are non-terminating non-recurring are as follows.   
 0.505005000500005000005…, 
 0.7207200720007200007200000…, 
 0.080080008000080000080000008… 
    

Question-8 :-  Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Solution :-
 5/7 = 0.714285
 9/11 = 0.81
 3 irrational numbers are as follows.   
 0.73073007300073000073…   
 0.75075007500075000075…   
 0.79079007900079000079…
   

Question-9 :-  Classify the following numbers as rational or irrational :
(i) √23, (ii) √225, (iii) 0.3796, (iv) 7.478478..., (v) 1.101001000100001...

Solution :-
(i)   √23 = 4.79583152331...
      As the decimal expansion of this number is non-terminating non-recurring, therefore, it is an irrational number. 

(ii)  √225 = 15
      It is a rational number as it can be represented in p/q form.

(iii) 0.3796
      As the decimal expansion of this number is terminating, therefore, it is a rational number.  

(iv)  7.478478... = 7.478 
      As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number. 
          
(v)   1.101001000100001...
      As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.
    
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