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Sagot :
To determine which of the given fractions are equivalent to [tex]\(\frac{1}{2}\)[/tex], we need to simplify each fraction and see if it simplifies to [tex]\(\frac{1}{2}\)[/tex].
Let's go through each fraction one by one:
1. [tex]\(\frac{3}{6}\)[/tex]:
- To simplify [tex]\(\frac{3}{6}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3.
- [tex]\(\frac{3 ÷ 3}{6 ÷ 3} = \frac{1}{2}\)[/tex]
- Therefore, [tex]\(\frac{3}{6}\)[/tex] is equivalent to [tex]\(\frac{1}{2}\)[/tex].
2. [tex]\(\frac{4}{5}\)[/tex]:
- This fraction is already in its simplest form. The numerator (4) and the denominator (5) have no common divisors other than 1.
- [tex]\(\frac{4}{5}\)[/tex] cannot be simplified to [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(\frac{4}{5}\)[/tex] is not equivalent to [tex]\(\frac{1}{2}\)[/tex].
3. [tex]\(\frac{6}{12}\)[/tex]:
- To simplify [tex]\(\frac{6}{12}\)[/tex], we divide both the numerator and the denominator by their GCD, which is 6.
- [tex]\(\frac{6 ÷ 6}{12 ÷ 6} = \frac{1}{2}\)[/tex]
- Therefore, [tex]\(\frac{6}{12}\)[/tex] is equivalent to [tex]\(\frac{1}{2}\)[/tex].
4. [tex]\(\frac{6}{7}\)[/tex]:
- This fraction is already in its simplest form. The numerator (6) and the denominator (7) have no common divisors other than 1.
- [tex]\(\frac{6}{7}\)[/tex] cannot be simplified to [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(\frac{6}{7}\)[/tex] is not equivalent to [tex]\(\frac{1}{2}\)[/tex].
5. [tex]\(\frac{5}{10}\)[/tex]:
- To simplify [tex]\(\frac{5}{10}\)[/tex], we divide both the numerator and the denominator by their GCD, which is 5.
- [tex]\(\frac{5 ÷ 5}{10 ÷ 5} = \frac{1}{2}\)[/tex]
- Therefore, [tex]\(\frac{5}{10}\)[/tex] is equivalent to [tex]\(\frac{1}{2}\)[/tex].
6. [tex]\(\frac{7}{8}\)[/tex]:
- This fraction is already in its simplest form. The numerator (7) and the denominator (8) have no common divisors other than 1.
- [tex]\(\frac{7}{8}\)[/tex] cannot be simplified to [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(\frac{7}{8}\)[/tex] is not equivalent to [tex]\(\frac{1}{2}\)[/tex].
Based on the analysis above, the fractions equivalent to [tex]\(\frac{1}{2}\)[/tex] are:
[tex]\[ \frac{3}{6}, \frac{6}{12}, \frac{5}{10} \][/tex]
Let's go through each fraction one by one:
1. [tex]\(\frac{3}{6}\)[/tex]:
- To simplify [tex]\(\frac{3}{6}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3.
- [tex]\(\frac{3 ÷ 3}{6 ÷ 3} = \frac{1}{2}\)[/tex]
- Therefore, [tex]\(\frac{3}{6}\)[/tex] is equivalent to [tex]\(\frac{1}{2}\)[/tex].
2. [tex]\(\frac{4}{5}\)[/tex]:
- This fraction is already in its simplest form. The numerator (4) and the denominator (5) have no common divisors other than 1.
- [tex]\(\frac{4}{5}\)[/tex] cannot be simplified to [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(\frac{4}{5}\)[/tex] is not equivalent to [tex]\(\frac{1}{2}\)[/tex].
3. [tex]\(\frac{6}{12}\)[/tex]:
- To simplify [tex]\(\frac{6}{12}\)[/tex], we divide both the numerator and the denominator by their GCD, which is 6.
- [tex]\(\frac{6 ÷ 6}{12 ÷ 6} = \frac{1}{2}\)[/tex]
- Therefore, [tex]\(\frac{6}{12}\)[/tex] is equivalent to [tex]\(\frac{1}{2}\)[/tex].
4. [tex]\(\frac{6}{7}\)[/tex]:
- This fraction is already in its simplest form. The numerator (6) and the denominator (7) have no common divisors other than 1.
- [tex]\(\frac{6}{7}\)[/tex] cannot be simplified to [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(\frac{6}{7}\)[/tex] is not equivalent to [tex]\(\frac{1}{2}\)[/tex].
5. [tex]\(\frac{5}{10}\)[/tex]:
- To simplify [tex]\(\frac{5}{10}\)[/tex], we divide both the numerator and the denominator by their GCD, which is 5.
- [tex]\(\frac{5 ÷ 5}{10 ÷ 5} = \frac{1}{2}\)[/tex]
- Therefore, [tex]\(\frac{5}{10}\)[/tex] is equivalent to [tex]\(\frac{1}{2}\)[/tex].
6. [tex]\(\frac{7}{8}\)[/tex]:
- This fraction is already in its simplest form. The numerator (7) and the denominator (8) have no common divisors other than 1.
- [tex]\(\frac{7}{8}\)[/tex] cannot be simplified to [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(\frac{7}{8}\)[/tex] is not equivalent to [tex]\(\frac{1}{2}\)[/tex].
Based on the analysis above, the fractions equivalent to [tex]\(\frac{1}{2}\)[/tex] are:
[tex]\[ \frac{3}{6}, \frac{6}{12}, \frac{5}{10} \][/tex]
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