Understanding Decimal Expansion Termination

The question asks about the decimal expansion of the fraction \(\frac{31}{2.5}\) and after how many decimal places it will terminate. A rational number has a terminating decimal expansion if and only if, when the fraction is written in its simplest form, the prime factors of the denominator are only 2s and 5s.

Converting to a Simple Fraction

First, we need to express the given expression as a simple fraction. The expression is \(\frac{31}{2.5}\).

We can convert the decimal 2.5 into a fraction:

\(2.5 = \frac{25}{10}\)

This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 5:

\(\frac{25}{10} = \frac{25 \div 5}{10 \div 5} = \frac{5}{2}\)

Now, substitute this back into the original expression:

\(\frac{31}{2.5} = \frac{31}{\frac{5}{2}}\)

To divide by a fraction, we multiply by its reciprocal:

\(\frac{31}{\frac{5}{2}} = 31 \times \frac{2}{5} = \frac{31 \times 2}{5} = \frac{62}{5}\)

So, the fraction is \(\frac{62}{5}\). This fraction is in its simplest form because 62 and 5 have no common factors other than 1.

Analyzing the Denominator for Termination

Now that the fraction is in its simplest form, \(\frac{62}{5}\), we look at the prime factors of the denominator, which is 5.

• The prime factors of 5 are just 5 itself.
• Since the only prime factor in the denominator is 5 (and not any other prime number), the decimal expansion of \(\frac{62}{5}\) will terminate.

Determining the Number of Decimal Places

To find out after how many decimal places the expansion terminates, we need to express the denominator in the form \(2^n \times 5^m\), where \(n\) and \(m\) are non-negative integers. The number of decimal places will be the maximum of \(n\) and \(m\), i.e., \(\max(n, m)\).

Our denominator is 5. We can write 5 as \(2^0 \times 5^1\).

• Here, the power of 2 is \(n=0\).
• The power of 5 is \(m=1\).
• The maximum of these powers is \(\max(0, 1) = 1\).

This indicates that the decimal expansion will terminate after 1 decimal place.

Alternatively, we can perform the division directly:

\(\frac{62}{5} = 12.4\)

The decimal representation is 12.4. This decimal has only one digit after the decimal point, which is 4. Therefore, the decimal expansion terminates after one decimal place.

Conclusion

The decimal expansion of \(\frac{31}{2.5}\), which simplifies to \(\frac{62}{5}\) or 12.4, terminates after one decimal place.

Revision Table: Decimal Expansion Termination

Additional Information: Types of Decimal Expansions

Rational numbers (numbers that can be expressed as a fraction \(\frac{p}{q}\), where p and q are integers and q is not zero) have decimal expansions that are either:

• Terminating: The decimal representation ends after a finite number of digits. This happens when the denominator of the fraction in simplest form has only 2s and/or 5s as prime factors. Example: \(\frac{1}{4} = 0.25\), \(\frac{3}{8} = 0.375\), \(\frac{1}{10} = 0.1\).
• Non-terminating and Repeating: The decimal representation continues infinitely with a repeating block of digits. This happens when the denominator of the fraction in simplest form contains prime factors other than 2 or 5. Example: \(\frac{1}{3} = 0.333...\), \(\frac{1}{7} = 0.142857142857...\), \(\frac{1}{6} = 0.1666...\).

Irrational numbers (like \(\sqrt{2}\) or \(\pi\)) have decimal expansions that are non-terminating and non-repeating.