Question

Which of the following numbers has non – terminating repeating decimal

expansion ?

(a) 29 / 3125
( b) 67 / 512
(c) 149 / 200
( d) none of these​

Answer 1

Answer:

A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result are irrational numbers. Pi is a non-terminating, non-repeating decimal.

Answer 2

(d) none of these is the correct answer.

Solution: A rational number in the form of p/q where q is not equal to 0, is terminating only when q is in the form of:

${2}^{m} \times {5}^{n}$

where m and n are whole numbers.

Checking option (a): 29/3125

Here, prime factorisation of 3125

= 5×5×5×5×5

$= {2}^{0} \times {5}^{5}$

Since it is expressed in the form, it is a terminating decimal.

Checking option (b): 67/512

Here, prime factorisation of 512

= 2×2×2×2×2×2×2×2×2

$= {2}^{9} \times {5}^{0}$

Since it is expressed in the form, it is a terminating decimal.

Checking option (c): 149/200

Here, prime factorisation of 200

= 2 ×2×2×5×5

$= {2}^{3} \times {5}^{2}$

Since it is expressed in the form, it is a terminating decimal.

Clearly, every option is a terminating decimal. None of them has non-terminating repeating decimal expansion.