Answer :
To determine which inequality is true among the given options, we will compare each pair of fractions. Here is the detailed step-by-step solution for each inequality:
### Inequality A: [tex]\(\frac{3}{4} < \frac{5}{7}\)[/tex]
To compare [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(4\)[/tex] and [tex]\(7\)[/tex] is [tex]\(28\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{3}{4} = \frac{3 \times 7}{4 \times 7} = \frac{21}{28} \][/tex]
[tex]\[ \frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28} \][/tex]
4. Compare the numerators: [tex]\(21\)[/tex] and [tex]\(20\)[/tex].
5. Since [tex]\(21 > 20\)[/tex], [tex]\(\frac{3}{4}\)[/tex] is actually greater than [tex]\(\frac{5}{7}\)[/tex].
Therefore, Inequality A is false.
### Inequality B: [tex]\(\frac{2}{3} > \frac{5}{6}\)[/tex]
To compare [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(3\)[/tex] and [tex]\(6\)[/tex] is [tex]\(6\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \][/tex]
[tex]\[ \frac{5}{6} = \frac{5}{6} \][/tex]
4. Compare the numerators: [tex]\(4\)[/tex] and [tex]\(5\)[/tex].
5. Since [tex]\(4 < 5\)[/tex], [tex]\(\frac{2}{3}\)[/tex] is actually less than [tex]\(\frac{5}{6}\)[/tex].
Therefore, Inequality B is false.
### Inequality C: [tex]\(\frac{5}{8} > \frac{6}{10}\)[/tex]
To compare [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{6}{10}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(8\)[/tex] and [tex]\(10\)[/tex] is [tex]\(40\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \][/tex]
[tex]\[ \frac{6}{10} = \frac{6 \times 4}{10 \times 4} = \frac{24}{40} \][/tex]
4. Compare the numerators: [tex]\(25\)[/tex] and [tex]\(24\)[/tex].
5. Since [tex]\(25 > 24\)[/tex], [tex]\(\frac{5}{8}\)[/tex] is greater than [tex]\(\frac{6}{10}\)[/tex].
Therefore, Inequality C is true.
### Inequality D: [tex]\(\frac{4}{5} < \frac{2}{9}\)[/tex]
To compare [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{2}{9}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(5\)[/tex] and [tex]\(9\)[/tex] is [tex]\(45\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45} \][/tex]
[tex]\[ \frac{2}{9} = \frac{2 \times 5}{9 \times 5} = \frac{10}{45} \][/tex]
4. Compare the numerators: [tex]\(36\)[/tex] and [tex]\(10\)[/tex].
5. Since [tex]\(36 > 10\)[/tex], [tex]\(\frac{4}{5}\)[/tex] is greater than [tex]\(\frac{2}{9}\)[/tex].
Therefore, Inequality D is false.
After evaluating all the inequalities, we find that the only true inequality among them is:
[tex]\[ \textbf{C:} \quad \frac{5}{8} > \frac{6}{10} \][/tex]
### Inequality A: [tex]\(\frac{3}{4} < \frac{5}{7}\)[/tex]
To compare [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(4\)[/tex] and [tex]\(7\)[/tex] is [tex]\(28\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{3}{4} = \frac{3 \times 7}{4 \times 7} = \frac{21}{28} \][/tex]
[tex]\[ \frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28} \][/tex]
4. Compare the numerators: [tex]\(21\)[/tex] and [tex]\(20\)[/tex].
5. Since [tex]\(21 > 20\)[/tex], [tex]\(\frac{3}{4}\)[/tex] is actually greater than [tex]\(\frac{5}{7}\)[/tex].
Therefore, Inequality A is false.
### Inequality B: [tex]\(\frac{2}{3} > \frac{5}{6}\)[/tex]
To compare [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(3\)[/tex] and [tex]\(6\)[/tex] is [tex]\(6\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \][/tex]
[tex]\[ \frac{5}{6} = \frac{5}{6} \][/tex]
4. Compare the numerators: [tex]\(4\)[/tex] and [tex]\(5\)[/tex].
5. Since [tex]\(4 < 5\)[/tex], [tex]\(\frac{2}{3}\)[/tex] is actually less than [tex]\(\frac{5}{6}\)[/tex].
Therefore, Inequality B is false.
### Inequality C: [tex]\(\frac{5}{8} > \frac{6}{10}\)[/tex]
To compare [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{6}{10}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(8\)[/tex] and [tex]\(10\)[/tex] is [tex]\(40\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \][/tex]
[tex]\[ \frac{6}{10} = \frac{6 \times 4}{10 \times 4} = \frac{24}{40} \][/tex]
4. Compare the numerators: [tex]\(25\)[/tex] and [tex]\(24\)[/tex].
5. Since [tex]\(25 > 24\)[/tex], [tex]\(\frac{5}{8}\)[/tex] is greater than [tex]\(\frac{6}{10}\)[/tex].
Therefore, Inequality C is true.
### Inequality D: [tex]\(\frac{4}{5} < \frac{2}{9}\)[/tex]
To compare [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{2}{9}\)[/tex]:
1. Convert both fractions to have a common denominator.
2. The common denominator for [tex]\(5\)[/tex] and [tex]\(9\)[/tex] is [tex]\(45\)[/tex].
3. Adjust the fractions:
[tex]\[ \frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45} \][/tex]
[tex]\[ \frac{2}{9} = \frac{2 \times 5}{9 \times 5} = \frac{10}{45} \][/tex]
4. Compare the numerators: [tex]\(36\)[/tex] and [tex]\(10\)[/tex].
5. Since [tex]\(36 > 10\)[/tex], [tex]\(\frac{4}{5}\)[/tex] is greater than [tex]\(\frac{2}{9}\)[/tex].
Therefore, Inequality D is false.
After evaluating all the inequalities, we find that the only true inequality among them is:
[tex]\[ \textbf{C:} \quad \frac{5}{8} > \frac{6}{10} \][/tex]
