Real Numbers Ex-1.4 Ch-1 [Solutions] NCERT Maths Class 10th

Exercise 1.4

Q1. 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) 23/23 × 52 (vii) 129/22 × 57 × 75 (viii) 6/15 (ix) 35/50 (x) 77/210

Answer

(i) 13/3125 Factorize the denominator we get 3125 =5 × 5 × 5 × 5 × 5 = 55 So denominator is in form of 5m so it is terminating .

(ii) 17/8 Factorize the denominator we get 8 =2 × 2 × 2 = 23 So denominator is in form of 2m so it is terminating .

(iii) 64/455 Factorize the denominator we get 455 =5 × 7 × 13 There are 7 and 13 also in denominator so denominator is not in form of 2m × 5n . so it is not terminating.

(iv) 15/1600 Factorize the denominator we get 1600 =2 × 2 × 2 ×2 × 2 × 2 × 5 × 5 = 26 × 52 so denominator is in form of 2m × 5n Hence it is terminating.

(v) 29/343 Factorize the denominator we get 343 = 7 × 7 × 7 = 73 There are 7 also in denominator so denominator is not in form of 2m × 5n Hence it is non-terminating.

(vi) 23/(23 × 52) Denominator is in form of 2m × 5n Hence it is terminating.

(vii) 129/(22 × 57 × 75 ) Denominator has 7 in denominator so denominator is not in form of 2m × 5n Hence it is none terminating.

(viii) 6/15 divide nominator and denominator both by 3 we get 2/5 Denominator is in form of 5m so it is terminating.

(ix) 35/50 divide denominator and nominator both by 5 we get 7/10 Factorize the denominator we get 10=2 × 5 So denominator is in form of 2m × 5n so it is terminating.

(x) 77/210 simplify it by dividing nominator and denominator both by 7 we get 11/30 Factorize the denominator we get 30=2 × 3 × 5 Denominator has 3 also in denominator so denominator is not in form of 2m × 5n Hence it is none terminating.

 

Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Answer

(i) 13/3125 = 13/55 = 13×25/55×25 = 416/105 = 0.00416 (ii) 17/8 = 17/23 = 17×53/23×53 = 17×53/103 = 2125/103 = 2.125

(iv) 15/1600 = 15/24×102 = 15×54/24×54×102 = 9375/106 = 0.009375

(vi) 23/2352 = 23×53×22/23 52×53×22 = 11500/105 = 0.115

(viii) 6/15 = 2/5 = 2×2/5×2 = 4/10 = 0.4

(ix) 35/50 = 7/10 = 0.7.

Q3. 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 p , q you say about the prime factors of q? (i) 43.123456789 (ii) 0.120120012000120000... (iii) 43.123456789

Answer:

(i) 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.

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