To determine which of the following statements is true, we will analyze each one step by step.
1. [tex]\(\frac{8}{16}=\frac{1}{4}\)[/tex]
Simplify [tex]\(\frac{8}{16}\)[/tex]:
[tex]\[
\frac{8}{16} = \frac{8 \div 8}{16 \div 8} = \frac{1}{2}
\][/tex]
This equality becomes [tex]\(\frac{1}{2} = \frac{1}{4}\)[/tex], which is false.
2. [tex]\(\frac{5}{6}>\frac{10}{12}\)[/tex]
Simplify [tex]\(\frac{10}{12}\)[/tex]:
[tex]\[
\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}
\][/tex]
This inequality becomes [tex]\(\frac{5}{6} > \frac{5}{6}\)[/tex], which is false, since they are equal.
3. [tex]\(\frac{3}{4}<\frac{4}{6}\)[/tex]
Simplify [tex]\(\frac{4}{6}\)[/tex]:
[tex]\[
\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}
\][/tex]
Now compare [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]:
Convert to a common denominator:
[tex]\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\][/tex]
[tex]\[
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\][/tex]
Comparing [tex]\(\frac{9}{12}\)[/tex] and [tex]\(\frac{8}{12}\)[/tex], we see that [tex]\(\frac{9}{12}\)[/tex] is not less than [tex]\(\frac{8}{12}\)[/tex], so the statement is false.
4. [tex]\(\frac{11}{15}<\frac{4}{5}\)[/tex]
Simplify [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{4}{5}
\][/tex]
Now compare [tex]\(\frac{11}{15}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex]:
Convert to a common denominator:
[tex]\[
\frac{11}{15} = \frac{11 \times 5}{15 \times 5} = \frac{55}{75}
\][/tex]
[tex]\[
\frac{4}{5} = \frac{4 \times 15}{5 \times 15} = \frac{60}{75}
\][/tex]
Comparing [tex]\(\frac{55}{75}\)[/tex] and [tex]\(\frac{60}{75}\)[/tex], we see that [tex]\(\frac{55}{75}\)[/tex] is indeed less than [tex]\(\frac{60}{75}\)[/tex], so the statement is true.
Based on this analysis, the true statement is:
[tex]\(\frac{11}{15} < \frac{4}{5}\)[/tex]
So, the correct answer is statement 4.
