To determine which of the given statements is true, we will evaluate each statement step-by-step.
1. First statement: [tex]\(\frac{5}{6} > \frac{10}{12}\)[/tex]
Let's compare the two fractions [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{10}{12}\)[/tex]:
[tex]\[
\frac{10}{12} = \frac{5 \times 2}{6 \times 2} = \frac{5}{6}
\][/tex]
So, [tex]\(\frac{5}{6} = \frac{10}{12}\)[/tex]. Thus, the statement [tex]\(\frac{5}{6} > \frac{10}{12}\)[/tex] is false.
2. Second statement: [tex]\(\frac{1}{4} > \frac{1}{3}\)[/tex]
Let's compare the two fractions [tex]\(\frac{1}{4}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex] by finding a common denominator. The lowest common denominator of 4 and 3 is 12.
[tex]\[
\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}
\][/tex]
[tex]\[
\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}
\][/tex]
Thus, [tex]\(\frac{3}{12} < \frac{4}{12}\)[/tex]. Therefore, [tex]\(\frac{1}{4} < \frac{1}{3}\)[/tex], making the statement [tex]\(\frac{1}{4} > \frac{1}{3}\)[/tex] false.
3. Third statement: [tex]\(\frac{6}{9} = \frac{1}{3}\)[/tex]
Let's simplify [tex]\(\frac{6}{9}\)[/tex]:
[tex]\[
\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}
\][/tex]
Comparing [tex]\(\frac{2}{3}\)[/tex] with [tex]\(\frac{1}{3}\)[/tex], we can see that they are not equal. Therefore, the statement [tex]\(\frac{6}{9} = \frac{1}{3}\)[/tex] is false.
4. Fourth statement: [tex]\(\frac{13}{15} > \frac{4}{5}\)[/tex]
Let's compare the two fractions [tex]\(\frac{13}{15}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex]:
First, convert [tex]\(\frac{4}{5}\)[/tex] to have a denominator of 15:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
Now, we compare [tex]\(\frac{13}{15}\)[/tex] and [tex]\(\frac{12}{15}\)[/tex]:
[tex]\[
\frac{13}{15} > \frac{12}{15}
\][/tex]
So, the statement [tex]\(\frac{13}{15} > \frac{4}{5}\)[/tex] is true.
Therefore, the only true statement is the fourth one: [tex]\(\frac{13}{15} > \frac{4}{5}\)[/tex].
