Answer :
To determine whether a fraction has a terminating or repeating decimal representation, we need to look at the denominator after simplifying the fraction.
A fraction in its simplest form will have a terminating decimal if its denominator (after simplification) is composed only of the prime factors 2 or 5. In other words, if the denominator can be expressed as [tex]\(2^n \times 5^m\)[/tex], where [tex]\(n\)[/tex] and [tex]\(m\)[/tex] are non-negative integers, then the fraction will have a terminating decimal. Otherwise, it will result in a repeating decimal.
Let's apply this rule to each of the given fractions:
(i) 25/64
- The fraction is already in its simplest form.
- The denominator 64 can be written as [tex]\(2^6\)[/tex].
- Since the denominator only has 2 as a prime factor, 25/64 has a terminating decimal representation.
(ii) 1/14
- The fraction is already in its simplest form.
- The denominator 14 can be written as [tex]\(2 \times 7\)[/tex].
- The presence of 7, which is not 2 or 5, means 1/14 has a repeating decimal representation.
(iii) 27/30
- First, simplify the fraction by finding the greatest common divisor (GCD) of 27 and 30, which is 3.
- Simplified fraction: 27/30 = 9/10.
- The simplified denominator 10 can be expressed as [tex]\(2 \times 5\)[/tex].
- Since the denominator consists only of the prime factors 2 and 5, 27/30 has a terminating decimal representation.
(iv) 1/30
- The fraction is already in its simplest form.
- The denominator 30 can be written as [tex]\(2 \times 3 \times 5\)[/tex].
- Since it contains 3, which is neither 2 nor 5, 1/30 has a repeating decimal representation.
Thus, the responses are as follows:
- 25/64: Terminating
- 1/14: Repeating
- 27/30: Terminating
- 1/30: Repeating
A fraction in its simplest form will have a terminating decimal if its denominator (after simplification) is composed only of the prime factors 2 or 5. In other words, if the denominator can be expressed as [tex]\(2^n \times 5^m\)[/tex], where [tex]\(n\)[/tex] and [tex]\(m\)[/tex] are non-negative integers, then the fraction will have a terminating decimal. Otherwise, it will result in a repeating decimal.
Let's apply this rule to each of the given fractions:
(i) 25/64
- The fraction is already in its simplest form.
- The denominator 64 can be written as [tex]\(2^6\)[/tex].
- Since the denominator only has 2 as a prime factor, 25/64 has a terminating decimal representation.
(ii) 1/14
- The fraction is already in its simplest form.
- The denominator 14 can be written as [tex]\(2 \times 7\)[/tex].
- The presence of 7, which is not 2 or 5, means 1/14 has a repeating decimal representation.
(iii) 27/30
- First, simplify the fraction by finding the greatest common divisor (GCD) of 27 and 30, which is 3.
- Simplified fraction: 27/30 = 9/10.
- The simplified denominator 10 can be expressed as [tex]\(2 \times 5\)[/tex].
- Since the denominator consists only of the prime factors 2 and 5, 27/30 has a terminating decimal representation.
(iv) 1/30
- The fraction is already in its simplest form.
- The denominator 30 can be written as [tex]\(2 \times 3 \times 5\)[/tex].
- Since it contains 3, which is neither 2 nor 5, 1/30 has a repeating decimal representation.
Thus, the responses are as follows:
- 25/64: Terminating
- 1/14: Repeating
- 27/30: Terminating
- 1/30: Repeating
