Identifying Terminating Decimal Representations

A rational number in the form of a fraction \(\frac{p}{q}\) (where \(p\) and \(q\) are integers and \(q \neq 0\)) has a terminating decimal representation if and only if the prime factorization of the denominator, \(q\), contains only the prime factors 2 and/or 5. If the prime factorization of the denominator contains any prime factors other than 2 or 5, the decimal representation will be non-terminating and repeating.

Let's examine each given mixed number option to determine if it has a terminating decimal representation.

Step-by-Step Analysis of Options

First, we convert each mixed number into an improper fraction. Then, we examine the prime factors of the denominator of the simplified fraction.

Option 1: \(2\frac{1}{3}\)

• Convert to improper fraction: \(2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\)
• Denominator is 3.
• Prime factors of 3: The only prime factor is 3.
• Since the denominator has a prime factor (3) other than 2 or 5, the decimal representation is non-terminating and repeating.

Option 2: \(3\frac{1}{7}\)

• Convert to improper fraction: \(3\frac{1}{7} = \frac{(3 \times 7) + 1}{7} = \frac{21 + 1}{7} = \frac{22}{7}\)
• Denominator is 7.
• Prime factors of 7: The only prime factor is 7.
• Since the denominator has a prime factor (7) other than 2 or 5, the decimal representation is non-terminating and repeating.

Option 3: \(1\frac{1}{5}\)

• Convert to improper fraction: \(1\frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{5 + 1}{5} = \frac{6}{5}\)
• Denominator is 5.
• Prime factors of 5: The only prime factor is 5.
• Since the denominator's prime factors consist only of 5 (which is allowed), the decimal representation is terminating.

Option 4: \(4\frac{1}{9}\)

• Convert to improper fraction: \(4\frac{1}{9} = \frac{(4 \times 9) + 1}{9} = \frac{36 + 1}{9} = \frac{37}{9}\)
• Denominator is 9.
• Prime factors of 9: \(9 = 3 \times 3\). The prime factors are 3 and 3.
• Since the denominator has a prime factor (3) other than 2 or 5, the decimal representation is non-terminating and repeating.

Based on the analysis, only \(1\frac{1}{5}\) (which is \(\frac{6}{5}\)) has a denominator whose prime factors are only 5. Therefore, \(1\frac{1}{5}\) has a terminating decimal representation.

To verify, \(1\frac{1}{5} = 1 + \frac{1}{5} = 1 + 0.2 = 1.2\), which is a terminating decimal.

Revision Table: Terminating vs. Non-Terminating Decimals

Additional Information: Rational Numbers and Decimal Expansions

A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, a numerator \(p\) and a non-zero denominator \(q\). Every rational number has a decimal expansion that is either terminating or non-terminating and repeating.

• Terminating Decimal: The decimal representation ends after a finite number of digits. Examples: 0.5, 1.25, 0.123.
• Non-terminating and Repeating Decimal: The decimal representation continues infinitely with a repeating block of digits. Examples: 0.333... (\(\frac{1}{3}\)), 0.142857142857... (\(\frac{1}{7}\)), 0.111... (\(\frac{1}{9}\)).

The key to distinguishing between these two types of decimal expansions for a rational number in its simplest fractional form \(\frac{p}{q}\) lies solely in the prime factorization of the denominator \(q\).