Without actually performing the long division, state whether the ratio

1.	Without actually performing the long division, state whether the rational number 64/455 have terminating or non-terminating repeating (recurring) decimal expansion.
Answer» Given rational number is 64/455 p/q is terminating if a) p and q are co-prime & b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers. Firstly, we check co-prime 64 = 2⁶ 455 = 5 × 7 × 13 64 and 455 have no common factors Therefore, 64 and 455 are co-prime. Now, we have to check that q is in the form of 2ⁿ5^m 455 = 5 × 7 × 13 So, denominator is not of the form 2ⁿ5^m Thus, 64455 is a non-terminating repeating decimal.

Without actually performing the long division, state whether the rational number 64/455 have terminating or non-terminating repeating (recurring) decimal expansion.

Answer»

Given rational number is 64/455

p/q is terminating if

a) p and q are co-prime &

b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.

Firstly, we check co-prime

64 = 2⁶

455 = 5 × 7 × 13

64 and 455 have no common factors

Therefore, 64 and 455 are co-prime.

Now, we have to check that q is in the form of 2ⁿ5^m

455 = 5 × 7 × 13

So, denominator is not of the form 2ⁿ5^m

Thus, 64455 is a non-terminating repeating decimal.