Sure, let's go through each comparison step by step to determine which ones are true and which are false.
1. [tex]\(\frac{1}{2} > \frac{1}{3}\)[/tex]:
- To compare these two fractions, we can find a common denominator. The common denominator between 2 and 3 is 6.
- [tex]\(\frac{1}{2} = \frac{3}{6}\)[/tex]
- [tex]\(\frac{1}{3} = \frac{2}{6}\)[/tex]
- Comparing [tex]\(\frac{3}{6} > \frac{2}{6}\)[/tex] is true, so [tex]\(\frac{1}{2} > \frac{1}{3}\)[/tex] is true.
2. [tex]\(\frac{2}{4} = \frac{3}{6}\)[/tex]:
- Simplify both fractions:
- [tex]\(\frac{2}{4} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{3}{6} = \frac{1}{2}\)[/tex]
- Since both simplify to [tex]\(\frac{1}{2}\)[/tex], they are indeed equal. Therefore, [tex]\(\frac{2}{4} = \frac{3}{6}\)[/tex] is true.
3. 1 < [tex]\(\frac{5}{10}\)[/tex]:
- Simplify [tex]\(\frac{5}{10}\)[/tex]:
- [tex]\(\frac{5}{10} = \frac{1}{2}\)[/tex]
- Comparing [tex]\(1\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
- [tex]\(1\)[/tex] is not less than [tex]\(\frac{1}{2}\)[/tex], so 1 < [tex]\(\frac{5}{10}\)[/tex] is false.
4. [tex]\(\frac{6}{12} > \frac{1}{3}\)[/tex]:
- Simplify both fractions:
- [tex]\(\frac{6}{12} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{1}{3}\)[/tex] remains [tex]\(\frac{1}{3}\)[/tex]
- Comparing [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]:
- [tex]\(\frac{1}{2}\)[/tex] is indeed greater than [tex]\(\frac{1}{3}\)[/tex], so [tex]\(\frac{6}{12} > \frac{1}{3}\)[/tex] is true.
5. [tex]\(\frac{2}{4} < \(\frac{6}{12}\)[/tex]:
- Simplify both fractions:
- [tex]\(\frac{2}{4} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{6}{12} = \frac{1}{2}\)[/tex]
- Comparing [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
- They are equal and thus [tex]\(\frac{1}{2}\)[/tex] is not less than [tex]\(\frac{1}{2}\)[/tex]. Therefore, [tex]\(\frac{2}{4} < \(\frac{6}{12}\)[/tex] is false.
So, the true comparisons among the given options are:
- [tex]\(\frac{1}{2} > \(\frac{1}{3}\)[/tex]
- [tex]\(\frac{2}{4} = \(\frac{3}{6}\)[/tex]
- [tex]\(\frac{6}{12} > \(\frac{1}{3}\)[/tex]
Thus, the final selection of true comparisons is [tex]\([ \text{True}, \text{True}, \text{False}, \text{True}, \text{False}]\)[/tex].
