Understanding Terminating and Recurring Decimals

A fraction can be converted into a decimal. The resulting decimal will either terminate (end) or recur (repeat a sequence of digits infinitely). The nature of the decimal form of a fraction depends on the prime factors of its denominator when the fraction is in its simplest form.

Rule for Terminating Decimals

A fraction \( \frac{p}{q} \) (where p and q are integers and q is not zero) will result in a terminating decimal if and only if the prime factors of the denominator q (in the simplified form of the fraction) are only 2s and/or 5s.

Rule for Recurring Decimals

A fraction \( \frac{p}{q} \) will result in a recurring decimal if the prime factors of the denominator q (in the simplified form of the fraction) include any prime factor other than 2 or 5.

Analyzing the Given Fractions

Let's examine each given fraction, simplify it to its lowest terms, and determine the prime factors of the denominator to see if it results in a terminating or recurring decimal.

• Fraction 1: 8/56

  Simplify the fraction:

  \( \frac{8}{56} = \frac{8 \div 8}{56 \div 8} = \frac{1}{7} \)

  The denominator is 7. The prime factor of 7 is 7. Since the prime factor (7) is not 2 or 5, this fraction will result in a recurring decimal.

• Fraction 2: 6/56

  Simplify the fraction:

  \( \frac{6}{56} = \frac{6 \div 2}{56 \div 2} = \frac{3}{28} \)

  The denominator is 28. Find the prime factors of 28:

  \( 28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7 \)

  The prime factors are 2 and 7. Since there is a prime factor (7) other than 2 or 5, this fraction will result in a recurring decimal.

• Fraction 3: 4/56

  Simplify the fraction:

  \( \frac{4}{56} = \frac{4 \div 4}{56 \div 4} = \frac{1}{14} \)

  The denominator is 14. Find the prime factors of 14:

  \( 14 = 2 \times 7 \)

  The prime factors are 2 and 7. Since there is a prime factor (7) other than 2 or 5, this fraction will result in a recurring decimal.

• Fraction 4: 7/56

  Simplify the fraction:

  \( \frac{7}{56} = \frac{7 \div 7}{56 \div 7} = \frac{1}{8} \)

  The denominator is 8. Find the prime factors of 8:

  \( 8 = 2 \times 4 = 2 \times 2 \times 2 = 2^3 \)

  The only prime factor is 2. Since the prime factors only include 2s (and no other primes like 3, 7, 11, etc.), this fraction will result in a terminating decimal.

Based on the analysis, the fraction 7/56 is the only one that results in a terminating decimal because the prime factors of its simplified denominator (8) are only 2s.

Revision Table: Fraction to Decimal Type

Additional Information on Decimal Expansions

Any rational number (a number that can be expressed as a fraction \( \frac{p}{q} \), where p and q are integers and \( q \neq 0 \)) will have a decimal expansion that is either terminating or recurring. Irrational numbers, like \( \sqrt{2} \) or \( \pi \), have decimal expansions that are non-terminating and non-recurring.

To convert a fraction to a decimal, you perform long division, dividing the numerator by the denominator. If the remainder becomes 0 at some point, the decimal terminates. If the remainder repeats, the decimal is recurring.