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
(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
