Refer each fraction below, a list of fractions and their equivalent fractions from the base list. Choose equivalent fraction from: 3/4, 2/4, 4/3, 2/3, 6/8, 3/6. Compare these pairs of fractions and insert >, < or =

To find an equivalent fraction for $\frac{1}{2}$, we can multiply the numerator and denominator by the same non-zero number.
For $\frac{2}{4}$:

$$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$$

This is an equivalent fraction.

$\frac{4}{4} = 1$, which is not equivalent to $\frac{1}{2}$.
$\frac{2}{3}$ is not equivalent to $\frac{1}{2}$.

The equivalent fraction is $\boxed{\mathbf{\frac{2}{4}}}$.

b) For $\frac{2}{3}$, choose from $\frac{3}{6}, \frac{1}{4}, \frac{3}{6}$.

To find an equivalent fraction for $\frac{2}{3}$ with a denominator of 6, we would multiply the numerator and denominator by 2:

$$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$$

Now, let's check the given options:
$\frac{3}{6}$ simplifies to $\frac{1}{2}$, which is not equal to $\frac{2}{3}$.
$\frac{1}{4}$ is not equal to $\frac{2}{3}$.
The option $\frac{3}{6}$ appears twice, and neither is equivalent to $\frac{2}{3}$.

Based on the given options, there is no equivalent fraction for $\frac{2}{3}$ in the list.

2. Compare these pairs of fractions and insert $>, <,$ or $=$:

a) Compare $\frac{2}{3}$ and $\frac{1}{3}$.

$$2 > 1$$

Therefore,

$$\frac{2}{3} \boxed{\mathbf{>}} \frac{1}{3}$$

b) Compare $\frac{2}{5}$ and $\frac{3}{5}$.

$$2 < 3$$

Therefore,

$$\frac{2}{5} \boxed{\mathbf{<}} \frac{3}{5}$$

c) Compare $\frac{1}{2}$ and $\frac{2}{4}$.

$$\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}$$

Therefore,

$$\frac{1}{2} \boxed{\mathbf{=}} \frac{2}{4}$$

d) Compare $\frac{2}{3}$ and $\frac{4}{6}$.

$$\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}$$

Therefore,

$$\frac{2}{3} \boxed{\mathbf{=}} \frac{4}{6}$$

e) Compare $\frac{2}{6}$ and $\frac{1}{3}$.

$$\frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}$$

Therefore,

$$\frac{2}{6} \boxed{\mathbf{=}} \frac{1}{3}$$