Terminating decimals

A fraction \(\dfrac{a}{b}\)(rational number) written in simplest form as can be written as a terminating decimal if and only if the prime factors of the denominator b are 2 or 5.

Example 3

Determine which of the following fractions can be written as a terminating decimal. Explain by giving the prime factorisation of the denominator.

1. \(\dfrac{1}{25}\)

Solution

\begin{align} 25 &= 5 × 5\\\\ &= 5^2\end{align} Hence \(\dfrac{1}{25}\) can be written as a terminating decimal. \begin{align}\dfrac{1}{25}×\dfrac{4}{4} &= \dfrac{4}{100}\\\\ &= 0.04\end{align}

1. \(\dfrac{1}{7}\)

Solution

7 = 7 × 1
This is a prime number which is neither 2 nor 5. Hence \(\dfrac{1}{7}\) cannot be written as a terminating decimal.

1. \(\dfrac{47}{64}\)

Solution

\begin{align}64 &= 2 × 2 × 2 × 2 × 2 × 2\\\\&= 2^6\end{align} Hence \(\dfrac{47}{64}\) can be written as a terminating decimal. Using the division algorithm,
\(\dfrac{47}{64} = 0.734375\)

1. \(\dfrac{1}{99}\)

Solution

\begin{align}99 &= 3 × 3 × 11\\\\ &= 3^2 × 11\end{align}

Both 3 and 11 are prime numbers, hence \(\dfrac{1}{99}\) cannot be written as a terminating decimal.