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Answer :
To determine how many of these fractions can be represented as terminating decimals, we need to understand a key concept: a fraction can be expressed as a terminating decimal if, after simplification, its denominator has only the prime factors 2 and/or 5. Let's analyze each fraction step by step:
1. Fraction: [tex]\(\frac{4}{25}\)[/tex]
- The prime factors of 25 are [tex]\(5 \times 5\)[/tex], which means the denominator is composed only of the prime factor 5.
- Therefore, [tex]\(\frac{4}{25}\)[/tex] can be expressed as a terminating decimal.
2. Fraction: [tex]\(\frac{73}{50}\)[/tex]
- The number 50 can be factored into [tex]\(2 \times 5^2\)[/tex]. This means the denominator factors are 2 and 5.
- Hence, [tex]\(\frac{73}{50}\)[/tex] can also be expressed as a terminating decimal.
3. Fraction: [tex]\(\frac{7}{30}\)[/tex]
- The number 30 can be factored into [tex]\(2 \times 3 \times 5\)[/tex]. Here, besides the factors 2 and 5, there is an extra factor of 3.
- This means [tex]\(\frac{7}{30}\)[/tex] cannot be expressed as a terminating decimal because of the presence of the factor 3.
4. Fraction: [tex]\(\frac{27}{30}\)[/tex]
- Like the previous fraction, the denominator 30 is [tex]\(2 \times 3 \times 5\)[/tex].
- The presence of the factor 3 in the denominator means [tex]\(\frac{27}{30}\)[/tex] cannot be expressed as a terminating decimal.
From this analysis, [tex]\(\frac{4}{25}\)[/tex] and [tex]\(\frac{73}{50}\)[/tex] can be expressed as terminating decimals. Therefore, 2 out of the 4 given fractions can be represented by terminating decimals.
1. Fraction: [tex]\(\frac{4}{25}\)[/tex]
- The prime factors of 25 are [tex]\(5 \times 5\)[/tex], which means the denominator is composed only of the prime factor 5.
- Therefore, [tex]\(\frac{4}{25}\)[/tex] can be expressed as a terminating decimal.
2. Fraction: [tex]\(\frac{73}{50}\)[/tex]
- The number 50 can be factored into [tex]\(2 \times 5^2\)[/tex]. This means the denominator factors are 2 and 5.
- Hence, [tex]\(\frac{73}{50}\)[/tex] can also be expressed as a terminating decimal.
3. Fraction: [tex]\(\frac{7}{30}\)[/tex]
- The number 30 can be factored into [tex]\(2 \times 3 \times 5\)[/tex]. Here, besides the factors 2 and 5, there is an extra factor of 3.
- This means [tex]\(\frac{7}{30}\)[/tex] cannot be expressed as a terminating decimal because of the presence of the factor 3.
4. Fraction: [tex]\(\frac{27}{30}\)[/tex]
- Like the previous fraction, the denominator 30 is [tex]\(2 \times 3 \times 5\)[/tex].
- The presence of the factor 3 in the denominator means [tex]\(\frac{27}{30}\)[/tex] cannot be expressed as a terminating decimal.
From this analysis, [tex]\(\frac{4}{25}\)[/tex] and [tex]\(\frac{73}{50}\)[/tex] can be expressed as terminating decimals. Therefore, 2 out of the 4 given fractions can be represented by terminating decimals.
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