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Sagot :
Sure, let's go through each fraction, determine if its decimal expansion is terminating, and find the decimal expansion step by step.
### 1. [tex]\(\frac{7}{8}\)[/tex]
Terminating: Yes, because the denominator [tex]\(8\)[/tex] can be factored into [tex]\(2^3\)[/tex].
Decimal Expansion: [tex]\(\frac{7}{8} = 0.875\)[/tex]
### 2. [tex]\(\frac{5}{10}\)[/tex]
Terminating: Yes, because the denominator [tex]\(10\)[/tex] can be factored into [tex]\(2 \times 5\)[/tex].
Decimal Expansion: [tex]\(\frac{5}{10} = 0.5\)[/tex]
### 3. [tex]\(\frac{2}{5}\)[/tex]
Terminating: Yes, because the denominator [tex]\(5\)[/tex] is already a factor of [tex]\(5\)[/tex].
Decimal Expansion: [tex]\(\frac{2}{5} = 0.4\)[/tex]
### 4. [tex]\(\frac{4}{25}\)[/tex]
Terminating: Yes, because the denominator [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex].
Decimal Expansion: [tex]\(\frac{4}{25} = 0.16\)[/tex]
### 5. [tex]\(\frac{15}{8}\)[/tex]
Terminating: Yes, because the denominator [tex]\(8\)[/tex] can be factored into [tex]\(2^3\)[/tex].
Decimal Expansion: [tex]\(\frac{15}{8} = 1.875\)[/tex]
### 6. [tex]\(\frac{15}{32}\)[/tex]
Terminating: Yes, because the denominator [tex]\(32\)[/tex] can be factored into [tex]\(2^5\)[/tex].
Decimal Expansion: [tex]\(\frac{15}{32} = 0.46875\)[/tex]
### 7. [tex]\(1 \frac{2}{25} = \frac{27}{25}\)[/tex]
Terminating: Yes, because the denominator [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex].
Decimal Expansion: [tex]\(\frac{27}{25} = 1.08\)[/tex]
### 8. [tex]\(2 \frac{7}{20} = \frac{47}{20}\)[/tex]
Terminating: Yes, because the denominator [tex]\(20\)[/tex] can be factored into [tex]\(2^2 \times 5\)[/tex].
Decimal Expansion: [tex]\(\frac{47}{20} = 2.35\)[/tex]
### 9. [tex]\(\frac{45}{96}\)[/tex]
Terminating: No, because the denominator [tex]\(96\)[/tex] factors into [tex]\(2^5 \times 3\)[/tex], and the presence of [tex]\(3\)[/tex] means it won't be terminating.
Decimal Expansion: [tex]\(\frac{45}{96} = 0.46875\)[/tex] (repeating part detected)
### 10. [tex]\(\frac{21}{14}\)[/tex]
Terminating: No, because the denominator [tex]\(14\)[/tex] factors into [tex]\(2 \times 7\)[/tex], and the presence of [tex]\(7\)[/tex] means it won't be terminating.
Decimal Expansion: [tex]\(\frac{21}{14} = 1.5\)[/tex] (repeating part detected)
### 11. [tex]\(\frac{49}{200}\)[/tex]
Terminating: Yes, because the denominator [tex]\(200\)[/tex] factors into [tex]\(2^3 \times 5^2\)[/tex].
Decimal Expansion: [tex]\(\frac{49}{200} = 0.245\)[/tex]
### 12. [tex]\(\frac{3}{750}\)[/tex]
Terminating: No, because the denominator [tex]\(750\)[/tex] factors into [tex]\(2 \times 3 \times 5^3\)[/tex], and the presence of [tex]\(3\)[/tex] means it won't be terminating.
Decimal Expansion: [tex]\(\frac{3}{750} = 0.004\)[/tex] (repeating part detected)
By examining each fraction's factors, we determined the nature of their decimal expansions and computed those expansions directly.
### 1. [tex]\(\frac{7}{8}\)[/tex]
Terminating: Yes, because the denominator [tex]\(8\)[/tex] can be factored into [tex]\(2^3\)[/tex].
Decimal Expansion: [tex]\(\frac{7}{8} = 0.875\)[/tex]
### 2. [tex]\(\frac{5}{10}\)[/tex]
Terminating: Yes, because the denominator [tex]\(10\)[/tex] can be factored into [tex]\(2 \times 5\)[/tex].
Decimal Expansion: [tex]\(\frac{5}{10} = 0.5\)[/tex]
### 3. [tex]\(\frac{2}{5}\)[/tex]
Terminating: Yes, because the denominator [tex]\(5\)[/tex] is already a factor of [tex]\(5\)[/tex].
Decimal Expansion: [tex]\(\frac{2}{5} = 0.4\)[/tex]
### 4. [tex]\(\frac{4}{25}\)[/tex]
Terminating: Yes, because the denominator [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex].
Decimal Expansion: [tex]\(\frac{4}{25} = 0.16\)[/tex]
### 5. [tex]\(\frac{15}{8}\)[/tex]
Terminating: Yes, because the denominator [tex]\(8\)[/tex] can be factored into [tex]\(2^3\)[/tex].
Decimal Expansion: [tex]\(\frac{15}{8} = 1.875\)[/tex]
### 6. [tex]\(\frac{15}{32}\)[/tex]
Terminating: Yes, because the denominator [tex]\(32\)[/tex] can be factored into [tex]\(2^5\)[/tex].
Decimal Expansion: [tex]\(\frac{15}{32} = 0.46875\)[/tex]
### 7. [tex]\(1 \frac{2}{25} = \frac{27}{25}\)[/tex]
Terminating: Yes, because the denominator [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex].
Decimal Expansion: [tex]\(\frac{27}{25} = 1.08\)[/tex]
### 8. [tex]\(2 \frac{7}{20} = \frac{47}{20}\)[/tex]
Terminating: Yes, because the denominator [tex]\(20\)[/tex] can be factored into [tex]\(2^2 \times 5\)[/tex].
Decimal Expansion: [tex]\(\frac{47}{20} = 2.35\)[/tex]
### 9. [tex]\(\frac{45}{96}\)[/tex]
Terminating: No, because the denominator [tex]\(96\)[/tex] factors into [tex]\(2^5 \times 3\)[/tex], and the presence of [tex]\(3\)[/tex] means it won't be terminating.
Decimal Expansion: [tex]\(\frac{45}{96} = 0.46875\)[/tex] (repeating part detected)
### 10. [tex]\(\frac{21}{14}\)[/tex]
Terminating: No, because the denominator [tex]\(14\)[/tex] factors into [tex]\(2 \times 7\)[/tex], and the presence of [tex]\(7\)[/tex] means it won't be terminating.
Decimal Expansion: [tex]\(\frac{21}{14} = 1.5\)[/tex] (repeating part detected)
### 11. [tex]\(\frac{49}{200}\)[/tex]
Terminating: Yes, because the denominator [tex]\(200\)[/tex] factors into [tex]\(2^3 \times 5^2\)[/tex].
Decimal Expansion: [tex]\(\frac{49}{200} = 0.245\)[/tex]
### 12. [tex]\(\frac{3}{750}\)[/tex]
Terminating: No, because the denominator [tex]\(750\)[/tex] factors into [tex]\(2 \times 3 \times 5^3\)[/tex], and the presence of [tex]\(3\)[/tex] means it won't be terminating.
Decimal Expansion: [tex]\(\frac{3}{750} = 0.004\)[/tex] (repeating part detected)
By examining each fraction's factors, we determined the nature of their decimal expansions and computed those expansions directly.
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