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Sagot :
To find which fractions are greater than [tex]\(\frac{2}{5}\)[/tex] and less than [tex]\(\frac{3}{5}\)[/tex], we need to evaluate each of the given fractions and compare them to these bounds.
Here are the given fractions to evaluate:
[tex]\[ \frac{1}{3}, \quad \frac{2}{6}, \quad \frac{2}{4}, \quad \frac{3}{6}, \quad \frac{5}{12}, \quad \frac{4}{10}, \quad \frac{7}{15} \][/tex]
Next, we convert [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] to decimal form for easier comparison:
[tex]\[ \frac{2}{5} = 0.4 \][/tex]
[tex]\[ \frac{3}{5} = 0.6 \][/tex]
We will now convert each of the given fractions to decimal form and compare it with [tex]\(0.4\)[/tex] and [tex]\(0.6\)[/tex].
1. [tex]\(\frac{1}{3} \approx 0.3333\)[/tex]
- [tex]\(0.3333 < 0.4\)[/tex]
- So, [tex]\(\frac{1}{3}\)[/tex] is not between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
2. [tex]\(\frac{2}{6}\)[/tex]
- Simplify: [tex]\(\frac{2}{6} = \frac{1}{3} = 0.3333\)[/tex]
- [tex]\(0.3333 < 0.4\)[/tex]
- So, [tex]\(\frac{2}{6}\)[/tex] is not between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
3. [tex]\(\frac{2}{4} = 0.5\)[/tex]
- [tex]\(0.4 < 0.5 < 0.6\)[/tex]
- So, [tex]\(\frac{2}{4}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
4. [tex]\(\frac{3}{6}\)[/tex]
- Simplify: [tex]\(\frac{3}{6} = \frac{1}{2} = 0.5\)[/tex]
- [tex]\(0.4 < 0.5 < 0.6\)[/tex]
- So, [tex]\(\frac{3}{6}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
5. [tex]\(\frac{5}{12} \approx 0.4167\)[/tex]
- [tex]\(0.4 < 0.4167 < 0.6\)[/tex]
- So, [tex]\(\frac{5}{12}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
6. [tex]\(\frac{4}{10} = 0.4\)[/tex]
- [tex]\(0.4 \leq 0.4\)[/tex]
- So, [tex]\(\frac{4}{10}\)[/tex] is not strictly greater than [tex]\(\frac{2}{5}\)[/tex]
7. [tex]\(\frac{7}{15} \approx 0.4667\)[/tex]
- [tex]\(0.4 < 0.4667 < 0.6\)[/tex]
- So, [tex]\(\frac{7}{15}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
Therefore, the fractions that are greater than [tex]\(\frac{2}{5}\)[/tex] and less than [tex]\(\frac{3}{5}\)[/tex] are:
[tex]\[ \frac{2}{4}, \quad \frac{3}{6}, \quad \frac{5}{12}, \quad \frac{7}{15} \][/tex]
Let's write them with their decimal equivalents as they were found:
[tex]\[ \left( \frac{2}{4}, 0.5 \right), \quad \left( \frac{3}{6}, 0.5 \right), \quad \left( \frac{5}{12}, 0.4167 \right), \quad \left( \frac{7}{15}, 0.4667 \right) \][/tex]
Here are the given fractions to evaluate:
[tex]\[ \frac{1}{3}, \quad \frac{2}{6}, \quad \frac{2}{4}, \quad \frac{3}{6}, \quad \frac{5}{12}, \quad \frac{4}{10}, \quad \frac{7}{15} \][/tex]
Next, we convert [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] to decimal form for easier comparison:
[tex]\[ \frac{2}{5} = 0.4 \][/tex]
[tex]\[ \frac{3}{5} = 0.6 \][/tex]
We will now convert each of the given fractions to decimal form and compare it with [tex]\(0.4\)[/tex] and [tex]\(0.6\)[/tex].
1. [tex]\(\frac{1}{3} \approx 0.3333\)[/tex]
- [tex]\(0.3333 < 0.4\)[/tex]
- So, [tex]\(\frac{1}{3}\)[/tex] is not between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
2. [tex]\(\frac{2}{6}\)[/tex]
- Simplify: [tex]\(\frac{2}{6} = \frac{1}{3} = 0.3333\)[/tex]
- [tex]\(0.3333 < 0.4\)[/tex]
- So, [tex]\(\frac{2}{6}\)[/tex] is not between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
3. [tex]\(\frac{2}{4} = 0.5\)[/tex]
- [tex]\(0.4 < 0.5 < 0.6\)[/tex]
- So, [tex]\(\frac{2}{4}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
4. [tex]\(\frac{3}{6}\)[/tex]
- Simplify: [tex]\(\frac{3}{6} = \frac{1}{2} = 0.5\)[/tex]
- [tex]\(0.4 < 0.5 < 0.6\)[/tex]
- So, [tex]\(\frac{3}{6}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
5. [tex]\(\frac{5}{12} \approx 0.4167\)[/tex]
- [tex]\(0.4 < 0.4167 < 0.6\)[/tex]
- So, [tex]\(\frac{5}{12}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
6. [tex]\(\frac{4}{10} = 0.4\)[/tex]
- [tex]\(0.4 \leq 0.4\)[/tex]
- So, [tex]\(\frac{4}{10}\)[/tex] is not strictly greater than [tex]\(\frac{2}{5}\)[/tex]
7. [tex]\(\frac{7}{15} \approx 0.4667\)[/tex]
- [tex]\(0.4 < 0.4667 < 0.6\)[/tex]
- So, [tex]\(\frac{7}{15}\)[/tex] is between [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
Therefore, the fractions that are greater than [tex]\(\frac{2}{5}\)[/tex] and less than [tex]\(\frac{3}{5}\)[/tex] are:
[tex]\[ \frac{2}{4}, \quad \frac{3}{6}, \quad \frac{5}{12}, \quad \frac{7}{15} \][/tex]
Let's write them with their decimal equivalents as they were found:
[tex]\[ \left( \frac{2}{4}, 0.5 \right), \quad \left( \frac{3}{6}, 0.5 \right), \quad \left( \frac{5}{12}, 0.4167 \right), \quad \left( \frac{7}{15}, 0.4667 \right) \][/tex]
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