To classify the decimal expansion of each fraction as a terminating decimal or a repeating decimal, we follow specific steps to determine the nature of the decimal.
### Step-by-Step Solution:
1. Check for Terminology:
- A terminating decimal is a decimal that ends or has a finite number of digits after the decimal point.
- A repeating decimal is a decimal that has one or more repeating digits after the decimal point infinitely.
2. Fraction Analysis:
- For a fraction to have a terminating decimal, the denominator (after simplifying the fraction to its lowest terms) must have only the prime factors 2 and/or 5.
- If the simplified denominator has any other prime factors, the decimal expansion will be repeating.
3. Given Fractions:
We will look at the fractions and check their decimal expansion nature.
- Fraction [tex]$\frac{56}{72}$[/tex]:
- Simplify the fraction: [tex]$\frac{56}{72} = \frac{7}{9}$[/tex].
- Denominator 9 has primes 3² which is not a combination of 2 or 5.
- Thus, [tex]$\frac{56}{72}$[/tex] results in a repeating decimal.
- Fraction [tex]$\frac{21}{22}$[/tex]:
- Simplify the fraction: it is already in the simplest form.
- Denominator 22 has primes 2 and 11, and 11 is not 2 or 5.
- Thus, [tex]$\frac{21}{22}$[/tex] results in a repeating decimal.
- Fraction [tex]$\frac{11}{121}$[/tex]:
- Simplify the fraction: [tex]$\frac{11}{121} = \frac{1}{11}$[/tex].
- Denominator 11 only has prime 11.
- Thus, [tex]$\frac{11}{121}$[/tex] results in a repeating decimal.
- Fraction [tex]$\frac{60}{120}$[/tex]:
- Simplify the fraction: [tex]$\frac{60}{120} = \frac{1}{2}$[/tex].
- Denominator 2 is a prime factor of 2.
- Thus, [tex]$\frac{60}{120}$[/tex] results in a terminating decimal.
### Final Classification:
Terminating Decimal:
- [tex]$\frac{60}{120}$[/tex]
Repeating Decimal:
- [tex]$\frac{56}{72}$[/tex]
- [tex]$\frac{21}{22}$[/tex]
- [tex]$\frac{11}{121}$[/tex]
