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in Real Numbers by (35.3k points)
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Write down the decimal expansion of the following  rational numbers and show whether these are terminating.

(i) 13/125

(ii) 14588/625

(iii) 59/500

1 Answer

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(i) 13/125 = 13/(5 x 5 x 5) = 13/53

= 13/(20 x 53)

Denominater is of the form of 2m x 5n.

Hence, 13/125 is the decimal terminating expansion

Representing in decimal form

13/125 = 13/53

Multiply (2)3 for making in power of 10 to denominator.

13/125 = (13 x 23)/(23 x 53

= (13 x 8)/(2 x 5)3 = 104/103

= 104/1000

= 0.104

Hence, decimal expansion of 13/125 is 0.104

(ii) 14588/625 = 14588/(5 x 5 x 5 x 5)

= 14588/(20 x 54)

denominator is of the form 2m x 5n.

So, 14588/625 is the decimal terminating expansion.

Representing in decimal form

14588/625 = 14588/54

Multiply 24 in  numerator and denominator

So, decimal expansion of given number is 23.3408.

(iii) 59/500 = 49/(2 x 2 x 5 x 5 x 5)

= 49/(22 x 53)

Since, denominator is in the form 2m x 5n.

So, 49/500 is decimal terminating expansion

Representing in the decimal form 

49/500 = 49/(22 x 53)

Multiply 2 in numerator and denominator

= (49 x 2)/(23 x 53) = 98/(2 x 5)3

= 98/103 = 98/1000 = 0.098

So, decimal expansion of given number is 0.098

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