To determine which ratios are equivalent to the ratio [tex]\(8:12\)[/tex], we will analyze each given ratio one by one.
Firstly, let's simplify the ratio [tex]\(8:12\)[/tex]:
1. [tex]\(8:12\)[/tex] can be simplified by dividing both numbers by their greatest common divisor (GCD), which is [tex]\(4\)[/tex].
[tex]\[
\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}
\][/tex]
Now, let's evaluate the given options to see if they simplify to the same ratio.
A) [tex]\(4:6\)[/tex]
1. Simplify [tex]\(4:6\)[/tex]:
[tex]\[
\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}
\][/tex]
This ratio simplifies to [tex]\(\frac{2}{3}\)[/tex], which is equivalent to [tex]\(8:12\)[/tex].
B) [tex]\(12:8\)[/tex]
1. Simplify [tex]\(12:8\)[/tex]:
[tex]\[
\frac{12}{8} = \frac{12 \div 4}{8 \div 4} = \frac{3}{2}
\][/tex]
This ratio simplifies to [tex]\(\frac{3}{2}\)[/tex], which is not equivalent to [tex]\(8:12\)[/tex].
C) [tex]\(16:20\)[/tex]
1. Simplify [tex]\(16:20\)[/tex]:
[tex]\[
\frac{16}{20} = \frac{16 \div 4}{20 \div 4} = \frac{4}{5}
\][/tex]
This ratio simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equivalent to [tex]\(8:12\)[/tex].
D) [tex]\(24:36\)[/tex]
1. Simplify [tex]\(24:36\)[/tex]:
[tex]\[
\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}
\][/tex]
This ratio simplifies to [tex]\(\frac{2}{3}\)[/tex], which is equivalent to [tex]\(8:12\)[/tex].
E) [tex]\(56:84\)[/tex]
1. Simplify [tex]\(56:84\)[/tex]:
[tex]\[
\frac{56}{84} = \frac{56 \div 28}{84 \div 28} = \frac{2}{3}
\][/tex]
This ratio simplifies to [tex]\(\frac{2}{3}\)[/tex], which is equivalent to [tex]\(8:12\)[/tex].
Therefore, the ratios that are equivalent to [tex]\(8:12\)[/tex] are:
[tex]\[
\boxed{A, \ D, \ E}
\][/tex]
