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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 Q.1

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This question is from class 10th real numbers. What is the different way for solving this question. Please help me for solving this question. Its also an important question of class 10th maths question.

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

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1. Note: If the denominator has only factors of 2 and 5 or in the form of 2m Ã—5n then it has terminating decimal expansion.

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

(i) 13/3125

Factorizing the denominator, we get,

3125 = 5 Ã— 5 Ã— 5 = 55

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

(ii) 17/8

Factorizing the denominator, we get,

8 = 2Ã—2Ã—2 = 23

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

(iii) 64/455

Factorizing the denominator, we get,

455 = 5Ã—7Ã—13

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

(iv) 15/ 1600

Factorizing the denominator, we get,

1600 = 2652

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

(v) 29/343

Factorizing the denominator, we get,

343 = 7Ã—7Ã—7 = 73 Since, the denominator is not in the form of 2m Ã— 5n thus 29/343 has a non-terminating decimal expansion.

(vi)23/(2352)

Clearly, the denominator is in the form of 2m Ã— 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 Ã— 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, thus, 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 Ã— 5n thus, 35/50 has a terminating decimal expansion.

(x) 77/210

77/210 = (7Ã— 11)/ (30 Ã— 7) = 11/30

Factorising the denominator, we get,

30 = 2 Ã— 3 Ã— 5

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

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