Let's examine each equation one by one and determine whether it is true or false.
1. Equation 1: [tex]\( 6 \div \frac{1}{3} > 16 \)[/tex]
When we divide 6 by [tex]\(\frac{1}{3}\)[/tex], it is equivalent to multiplying 6 by 3:
[tex]\[
6 \div \frac{1}{3} = 6 \times 3 = 18
\][/tex]
So, we have:
[tex]\[
18 > 16
\][/tex]
This statement is true.
2. Equation 2: [tex]\( \frac{1}{4} \div 3 = \frac{3}{12} \)[/tex]
Dividing [tex]\(\frac{1}{4}\)[/tex] by 3 means multiplying [tex]\(\frac{1}{4}\)[/tex] by the reciprocal of 3, which is [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{1}{4} \div 3 = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}
\][/tex]
Now, let's examine the right-hand side:
[tex]\[
\frac{3}{12} = \frac{1}{4}
\][/tex]
Since [tex]\(\frac{1}{12} \neq \frac{1}{4}\)[/tex], this statement is false.
3. Equation 3: [tex]\( 12 \div \frac{1}{6} < 80 \)[/tex]
Dividing 12 by [tex]\(\frac{1}{6}\)[/tex] is equivalent to multiplying 12 by 6:
[tex]\[
12 \div \frac{1}{6} = 12 \times 6 = 72
\][/tex]
So, we have:
[tex]\[
72 < 80
\][/tex]
This statement is true.
4. Equation 4: [tex]\( \frac{1}{5} \div 2 > 9 \)[/tex]
Dividing [tex]\(\frac{1}{5}\)[/tex] by 2 is equivalent to multiplying [tex]\(\frac{1}{5}\)[/tex] by the reciprocal of 2, which is [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{5} \div 2 = \frac{1}{5} \times \frac{1}{2} = \frac{1}{10}
\][/tex]
So, we have:
[tex]\[
\frac{1}{10} > 9
\][/tex]
Since [tex]\(\frac{1}{10}\)[/tex] is much less than 9, this statement is false.
Summarizing our findings:
[tex]\[
\begin{tabular}{|l|c|c|}
\hline & . True & False \\
\hline $6 \div \frac{1}{3} > 16$ & \bigcirc & \\
\hline $\frac{1}{4} \div 3 = \frac{3}{12}$ & & \bigcirc \\
\hline $12 \div \frac{1}{6} < 80$ & \bigcirc & \\
\hline $\frac{1}{5} \div 2 > 9$ & & \bigcirc \\
\hline
\end{tabular}
\][/tex]
