State whether the following rational numbers

Without actually performing the king 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) 23/75
(iv) 6/15
(v) 1258/625
(vi) 77/210

Solution:

We know that, if the denominator of a fraction has only 2 or 5 or both factors, it is a terminating decimal otherwise it is non-terminating repeating decimals.
(i) 13/3125
Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 17
3125 = 5 × 5 × 5 × 5 × 5
Prime factor of 3125 = 5, 5, 5, 5, 5 [i.e., in the form of 2ⁿ, 5ⁿ]
It is a terminating decimal.
(ii) 17/8

8 = 2 × 2 × 2
Prime factor of 8 = 2, 2, 2 [i.e., in the form of 2ⁿ, 5ⁿ]
It is a terminating decimal.
(iii) 23/75
Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 19
75 = 3 × 5 × 5
Prime factor of 75 = 3, 5, 5
It is a non-terminating repeating decimal.
(iv) 6/15
Let us divide both numerator and denominator by 3
6/15 = (6 ÷ 3) / (15 ÷ 3)
= 2/5
Since the denominator is 5.
It is a terminating decimal.
(v) 1258/625
Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 20
625 = 5 × 5 × 5 × 5
Prime factor of 625 = 5, 5, 5, 5 [i.e., in the form of 2ⁿ, 5ⁿ]
It is a terminating decimal.
(vi) 77/210
Let us divide both numerator and denominator by 7
77/210 = (77 ÷ 7) / (210 ÷ 7)
= 11/30
Rational and Irrational Numbers Class 9 ICSE ML Aggarwal img 21
30 = 2 × 3 × 5
Prime factor of 30 = 2, 3, 5
It is a non-terminating repeating decimal.