Question-1 :- Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) 13/3125,  (ii) 17/8, (iii) 64/455, (iv) 15/1600,  (v) 29/343, (vi) 23/2³5², (vii) 129/2²5⁷7⁵,  (viii) 6/15, (ix) 35/50, (x) 77/210

(i) 13/3125
    3125 = 55 
    The denominator is of the form 5m.
    Hence, the decimal expansion of 13/3125 is terminating.
    

(ii) 17/8
     17/8 = 2³
     The denominator is of the form 2m.
     Hence, the decimal expansion of 17/8 is terminating.
    

(iii) 64/455
     455 = 5 x 7 x 13
     Since the denominator is not in the form 2m × 5n , and it also contains 7 and 13 as its factors, 
     its decimal expansion will be non-terminating repeating.
    

(iv) 15/1600
     1600 = 26 × 5²
     The denominator is of the form 2m × 5n.
     Hence, the decimal expansion of 15/1600 is terminating.
    

(v)  29/343
     343 = 7³
     Since the denominator is not in the form 2m × 5n , and it also contains 7 as its factors, 
     its decimal expansion will be non-terminating repeating.
    

(vi) 23/2³5²
     The denominator is of the form 2m × 5n.
     Hence, the decimal expansion of 23/2³5² is terminating.
    

(vii) 129/2²5775
     Since the denominator is not in the form 2m × 5n , and it also contains 7 as its factors, 
     its decimal expansion will be non-terminating repeating.
    

(viii) 6/15
     6/15 = (2 x 3)/(3 x 5) = 2/5
     The denominator is of the form 5m.
     Hence, the decimal expansion of 6/15 is terminating.
    

(ix) 35/50
     35/50 = (5 x 7)/(2 x 5 x 5) = 7/(2 x 5)
     The denominator is of the form 2m × 5n.
     Hence, the decimal expansion of 35/50 is terminating.
    

(x)  77/210
     77/210 = (11 x 7)/(30 x 7) = 11/30 = 11/(2 x 3 x 5)
     Since the denominator is not in the form 2m × 5n , and it also contains 3 as its factors, 
     its decimal expansion will be non-terminating repeating.
  

Question-2 :-  Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
(i) 13/3125,  (ii) 17/8, (iv) 15/1600,  (vi) 23/2³5², (viii) 6/15, (ix) 35/50

Question-3 :- The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , what can you say about the prime factor of q? (i) 43.123456789,  (ii) 0.120120012000120000…,  (iii) 43.123456789

(i) 43.123456789
    Since this number has a terminating decimal expansion, it is a rational
    number of the form p/q and q is of the form 2m × 5n 
    i.e., the prime factors of q will be either 2 or 5 or both. 
    

(ii) 0.120120012000120000…
     The decimal expansion is neither terminating nor recurring. 
     Therefore, the given number is an irrational number.
    

(iii) 43.123456789
      Since the decimal expansion is non-terminating recurring, the given
      number is a rational number of the form p/q and q is not of the form 2m × 5n 
      i.e., the prime factors of q will also have a factor other than 2 or 5.