Understanding Terminating Decimals

A fraction can be expressed as a terminating decimal if and only if the prime factorization of its denominator contains only the primes 2 and 5. The number of decimal places after which the decimal expansion terminates is determined by the highest power of 2 or 5 in the prime factorization of the denominator.

Let the fraction in its simplest form be \(\frac{p}{q}\). If the prime factorization of the denominator \(q\) is \(q = 2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers, then the decimal expansion of \(\frac{p}{q}\) will terminate after \(\max(m, n)\) decimal places.

Analyzing the Fraction \(\frac{27}{25}\)

We are given the fraction \(\frac{27}{25}\). To find out after how many decimal places its decimal expansion will terminate, we need to examine the denominator, which is 25.

Step 1: Prime Factorization of the Denominator

The denominator is 25. We find the prime factorization of 25:

• Divide 25 by the smallest prime number, 5: \(25 \div 5 = 5\)
• Divide 5 by 5: \(5 \div 5 = 1\)

So, the prime factorization of 25 is \(5 \times 5\), which can be written as \(5^2\).

Step 2: Expressing the Denominator in the Form \(2^m \times 5^n\)

The prime factorization of the denominator 25 is \(5^2\). To express this in the form \(2^m \times 5^n\), we can write it as \(2^0 \times 5^2\). Here, the power of 2 is \(m=0\) and the power of 5 is \(n=2\).

The denominator is \(25 = 2^0 \times 5^2\).

Step 3: Determining the Number of Decimal Places

The number of decimal places after which the decimal expansion terminates is the maximum of the powers of 2 and 5 in the denominator's prime factorization. In this case, \(m=0\) and \(n=2\).

We need to find \(\max(m, n) = \max(0, 2)\).

\(\max(0, 2) = 2\).

Therefore, the decimal expansion of \(\frac{27}{25}\) will terminate after 2 decimal places.

Verification by Decimal Conversion

We can also convert the fraction \(\frac{27}{25}\) to a decimal to verify our result.

To convert \(\frac{27}{25}\) to a decimal, we can perform division or multiply the numerator and denominator by a factor that makes the denominator a power of 10.

Using multiplication:

\(\frac{27}{25} = \frac{27 \times 4}{25 \times 4} = \frac{108}{100}\)

Converting \(\frac{108}{100}\) to a decimal:

\(\frac{108}{100} = 1.08\)

The decimal representation is 1.08, which has exactly two digits after the decimal point. This confirms that the decimal expansion terminates after two decimal places.

Conclusion on Decimal Termination

Based on the prime factorization of the denominator 25, which is \(2^0 \times 5^2\), the maximum power is 2. This indicates that the decimal expansion of \(\frac{27}{25}\) terminates after two decimal places.

Revision Table: Decimal Expansions

Additional Information: Prime Factorization and Terminating Decimals

The concept of terminating decimals is closely linked to the structure of the decimal number system, which is based on powers of 10. A fraction \(\frac{p}{q}\) (in simplest form) can be written with a denominator that is a power of 10 (\(10^k\)) if and only if \(q\) is a factor of some power of 10.

Since \(10 = 2 \times 5\), any power of 10, say \(10^k\), can be written as \((2 \times 5)^k = 2^k \times 5^k\). For \(q\) to be a factor of \(10^k\), the prime factors of \(q\) must only be 2 and 5, and the powers of these primes in \(q\) must be less than or equal to \(k\).

If \(q = 2^m \times 5^n\), we need to find the smallest \(k\) such that \(2^m \times 5^n\) divides \(2^k \times 5^k\). This requires \(m \le k\) and \(n \le k\), so the smallest such \(k\) is \(\max(m, n)\). When the denominator is transformed into \(10^{\max(m, n)}\), the number of zeros in the denominator is \(\max(m, n)\), which directly corresponds to the number of decimal places.

For example, \(\frac{3}{8} = \frac{3}{2^3}\). Here \(m=3, n=0\). \(\max(3,0)=3\). We multiply to get denominator \(10^3\): \(\frac{3}{2^3} = \frac{3 \times 5^3}{2^3 \times 5^3} = \frac{3 \times 125}{10^3} = \frac{375}{1000} = 0.375\). It terminates after 3 places.