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CLASS 10 Chapter 1 – Real Number Exercise 1.4
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

Solutions:

Note: If the denominator has only factors of 2 and 5 or in the form of 2m ×5n, then it has a terminating decimal expansion.

If the denominator has factors other than 2 and 5, then it has a non-terminating decimal expansion.

(i) 13/3125

Factorising the denominator, we get

3125 = 5 × 5 × 5 = 55

Since the denominator has only 5 as its factor, 13/3125 has a terminating decimal expansion.

(ii) 17/8

Factorising the denominator, we get

8 = 2×2×2 = 23

Since the denominator has only 2 as its factor, 17/8 has a terminating decimal expansion.

(iii) 64/455

Factorising the denominator, we get

455 = 5×7×13

Since the denominator is not in the form of 2m × 5n, 64/455 has a non-terminating decimal expansion.

(iv) 15/ 1600

Factorising the denominator, we get

1600 = 26×52

Since the denominator is in the form of 2m × 5n, 15/1600 has a terminating decimal expansion.

(v) 29/343

Factorising the denominator, we get

343 = 7×7×7 = 73

Since the denominator is not in the form of 2m × 5n, 29/343 has a non-terminating decimal expansion.

(vi)23/(2352)

Clearly, the denominator is in the form of 2m × 5n.

Hence, 23/ (2352) has a terminating decimal expansion.

(vii) 129/(225775)

As you can see, the denominator is not in the form of 2m × 5n.

Hence, 129/ (225775) has a non-terminating decimal expansion.

(viii) 6/15

6/15 = 2/5

Since the denominator has only 5 as its factor, 6/15 has a terminating decimal expansion.

(ix) 35/50

35/50 = 7/10

Factorising the denominator, we get

10 = 2 × 5

Since the denominator is in the form of 2m × 5n, 35/50 has a terminating decimal expansion.

(x) 77/210

77/210 = (7× 11)/ (30 × 7) = 11/30

Factorising the denominator, we get

30 = 2 × 3 × 5

As you can see, the denominator is not in the form of 2m × 5n. Hence, 77/210 has a non-terminating decimal expansion.