Question

Find which of the following rational numbers have terminating decimals. (1) 23/125 (2) 43/50 (3) 32/45
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Answer 1

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Let x = p /q  be a rational number, such that the prime factorisation of q is of the form 2^n *5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates. condition 1

: Let x = p /q be a rational number, such that the prime factorisation  of q is not of the form 2^n  *5^m, where n, m are non-negative integers. Then, x has a

decimal expansion which is non-terminating repeating  condition 2

23/125 = 23/5*5*5= 23/5³ it  terminates according to condition 1

43/50 = 43/5*5*2=43/5²*2 it  terminates according to condition 1

32/45=32/3*3*5=32/3²*5 it does not terminates according to condition 2

Answer 2

Answer:

The rational no having denominator 3,7,9,11,13,17,23,27.............. and multiple of these number will have non terminating decimal .

(1)

23/125 the denominator of this rational number is not having these above number and multiple of these number, so this will have terminating decimal.

(3)

32/45 the denominator of this rational number is having these above number multiple of 9, so this will have non terminating decimal.

(2)

43/50 the denominator of this rational number is not having these above number and multiple of these number, so this will have terminating decimal.

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