Understanding the Decimal Expansion of \(\frac{6}{15}\)

To determine the nature of the decimal expansion of a rational number like \(\frac{6}{15}\), we first need to simplify the fraction to its lowest terms.

The fraction is given as \(\frac{6}{15}\). We can find the greatest common divisor (GCD) of the numerator (6) and the denominator (15). The factors of 6 are 1, 2, 3, 6. The factors of 15 are 1, 3, 5, 15. The GCD of 6 and 15 is 3.

Now, we divide both the numerator and the denominator by their GCD:

\[ \frac{6 \div 3}{15 \div 3} = \frac{2}{5} \]

So, the simplified form of \(\frac{6}{15}\) is \(\frac{2}{5}\). Now we need to find the decimal expansion of \(\frac{2}{5}\).

A rational number \(\frac{p}{q}\) (where p and q are integers, \(q \neq 0\), and p and q are coprime) has a terminating decimal expansion if and only if the prime factors of the denominator, q, are only 2s and/or 5s. If the prime factors of the denominator include any prime number other than 2 or 5, the decimal expansion will be non-terminating and recurring.

Prime Factors of the Denominator

In the simplified fraction \(\frac{2}{5}\), the denominator is 5.

Let's find the prime factors of the denominator 5:

• The prime factors of 5 are just 5.

Since the prime factors of the denominator (5) contain only the prime number 5, according to the rule, the decimal expansion of \(\frac{2}{5}\) (and thus \(\frac{6}{15}\)) will be a terminating decimal.

Performing the Division

We can also perform the division to confirm:

\[ \frac{2}{5} = 0.4 \]

The decimal representation 0.4 ends after one digit, which means it is a terminating decimal expansion. This confirms our conclusion based on the prime factors of the denominator.

Therefore, the decimal expansion of \(\frac{6}{15}\) is terminating. It is not a non-terminating recurring decimal or a non-terminating non-recurring decimal.

To summarise, the nature of the decimal expansion is determined by the prime factors of the denominator of the simplified fraction. For \(\frac{6}{15}\), simplifying gives \(\frac{2}{5}\). The denominator 5 has only 5 as a prime factor. This leads to a terminating decimal.