To compare the fractions [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{4}{6}\)[/tex] using [tex]\(>\)[/tex], [tex]\(=\)[/tex], or [tex]\(<\)[/tex], follow these steps:
1. Simplify the fractions if possible:
- The fraction [tex]\(\frac{3}{7}\)[/tex] is already in its simplest form since 3 and 7 have no common factors other than 1.
- The fraction [tex]\(\frac{4}{6}\)[/tex] can be simplified:
[tex]\[
\frac{4 \div 2}{6 \div 2} = \frac{2}{3}
\][/tex]
So instead of comparing [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{4}{6}\)[/tex], we will compare [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex].
2. Find a common denominator for the two fractions to compare them:
- The denominators are 7 and 3. The least common multiple (LCM) of 7 and 3 is 21.
- Convert each fraction to have the same denominator:
[tex]\[
\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}
\][/tex]
[tex]\[
\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}
\][/tex]
3. Compare the fractions:
- Now we compare [tex]\(\frac{9}{21}\)[/tex] and [tex]\(\frac{14}{21}\)[/tex].
- Since the denominators are the same, compare the numerators directly:
[tex]\[
9 \, \text{and} \, 14
\][/tex]
Clearly, [tex]\(9 < 14\)[/tex].
Therefore, [tex]\(\frac{9}{21} < \frac{14}{21}\)[/tex], which means:
[tex]\[
\frac{3}{7} < \frac{2}{3}
\][/tex]
4. Conclude the comparison:
- So the comparison of [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{4}{6}\)[/tex] (which simplifies to [tex]\(\frac{2}{3}\)[/tex]) yields:
[tex]\[
\frac{3}{7} < \frac{4}{6}
\][/tex]
Thus, the correct answer is [tex]\(\frac{3}{7} < \frac{4}{6}\)[/tex].
