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 :​

Answer 1

Answer:

take out the lcm of the denominator result as

lcm of 60 is 2,2,5,3

if the lcm of denominator is any number except 2 and 5 then the decimal will be non terminating

as the number 3

it is non terminating

Answer 2

Answer:

Theorem: Let x=qp be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which terminates.

(i) 312513

Factorise the denominator, we get

3125=5×5×5×5×5=55

So, denominator is in form of 5m so, 312513 is terminating.

(ii) 817

Factorise the denominator, we get

8=2×2×2=23

So, denominator is in form of 2n so, 817 is terminating.

(iii) 45564

Factorise the denominator, we get

455=5×7×13

So, denominator is not in form of 2n5m so, 45564 is not terminating.

(iv) 160015

Factorise the denominator, we get

1600=2×2×2×2×2×2×5×5=2652

So, denominator is in form of 2n5m so, 160015