Without actually performing the long division, state whether the ratio

1.	Without actually performing the long division, state whether the rational number 27/8 have terminating or non-terminating repeating (recurring) decimal expansion.
Answer» Given rational number is 27/8 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 27 = 3 × 3 × 3 8 = 2 × 2 × 2 27 and 8 have no common factors Therefore, 27 and 8 are co-prime. Now, we have to check that q is in the form of 2ⁿ5^m 8 = 2³ = 1 × 2³ = 5⁰ × 2³ So, denominator is of the form 2ⁿ5^m where n = 3 and m = 0 Thus, 27/8 is a terminating decimal.

Without actually performing the long division, state whether the rational number 27/8 have terminating or non-terminating repeating (recurring) decimal expansion.

Answer»

Given rational number is 27/8

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

27 = 3 × 3 × 3

8 = 2 × 2 × 2

27 and 8 have no common factors

Therefore, 27 and 8 are co-prime.

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

8 = 2³

= 1 × 2³

= 5⁰ × 2³

So, denominator is of the form 2ⁿ5^m where n = 3 and m = 0

Thus, 27/8 is a terminating decimal.