Without performing division, determine whether the decimal expansion of is terminating or non-terminating. If it terminates, state the number of decimal places.

Question

Homi · Accepted Answer

Understand the Question

A rational number $\dfrac{p}{q}$ in its simplest form (where $p$ and $q$ are co-prime) has a terminating decimal expansion if the prime factorisation of $q$ is of the form $2^n \times 5^m$ , where $n$ and $m$ are non-negative integers.
If the denominator has any prime factor other than $2$ or $5$ , the decimal expansion is non-terminating repeating.
The number of decimal places after which the expansion terminates is given by $\max(n, m)$ .

Step 1 · Prime factorise the denominator

Given fraction $\dfrac{18}{125}$

Since $\text{HCF}(18, 125) = 1$ , the fraction is already in its simplest form $\dfrac{p}{q}$ .

Prime factorising the denominator $q = 125$ $125 = 5 \times 5 \times 5 = 5^3 = 2^0 \times 5^3$

Step 2 · Check condition and determine decimal places

The denominator is of the form $2^n \times 5^m$ , where $n = 0$ and $m = 3$ are non-negative integers.

Therefore, the decimal expansion of $\dfrac{18}{125}$ is terminating.

The number of decimal places is $\max(n, m) = \max(0, 3) = 3$ .

$\dfrac{18}{125} = \dfrac{18 \times 2^3}{5^3 \times 2^3} = \dfrac{18 \times 8}{10^3} = \dfrac{144}{1000} = 0.144$

Answer

The decimal expansion is terminating and terminates after $3$ decimal places.

Common Mistakes

Simplest Form Check: Forgetting to check if the numerator and denominator are co-prime before factorising the denominator.
Decimal Places Rule: Adding the powers ( $n + m$ ) instead of taking the maximum $\max(n, m)$ to find the number of decimal places.

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