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

Foundation

Mathematics Foundation

HCF and LCM

Recurring Decimal Number

Rational Numbers

Question

Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $\frac{17}{8}$ .

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Given, rational number is $\frac{17}{8}$ .

We know,

\frac{p}{q}

is terminating if:

a) p and q are co-prime and

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

Firstly, we check whether the numerator and denominator are co-prime,

17 = 17 \times 1

8 = 2 \times 2 \times 2

\Rightarrow 17

and 8 have no common factors.

Therefore, 17 and 8 are co-prime.

Now, we check whether q is in the form of 2ⁿ× 5^m,

denominator 8 = 2³ = 1 × 2³ = 5⁰× 2³

Since the denominator is of the form 2ⁿ× 5^m where n = 3 and m = 0,

thus,

\frac{17}{8}

has a terminating decimal expansion.