Let the total capacity of the jar be $C$ milliliters.

Initially, the jar is $\frac{1}{3}$ full, so it contains $\frac{C}{3}$ milliliters of juice.

After adding 500 milliliters, the jar is $\frac{5}{8}$ full, meaning it contains $\frac{5C}{8}$ milliliters of juice.

We can set up the equation:

$\frac{C}{3} + 500 = \frac{5C}{8}$

To solve for $C$, first eliminate the fractions by multiplying the entire equation by the least common multiple of 3 and 8, which is 24:

$24 \times \left(\frac{C}{3}\right) + 24 \times 500 = 24 \times \left(\frac{5C}{8}\right)$

This simplifies to:

$8C + 12000 = 15C$

Next, subtract $8C$ from both sides:

$12000 = 7C$

Finally, solve for $C$:

$C = \frac{12000}{7} \approx 1714.29 \, \text{milliliters}$

So, the total capacity of the jar is approximately 1714.29 milliliters.

Would you like further details on this, or do you have any other questions?

Here are 5 related questions:

1. How would the equation change if the jar were initially 1/4 full instead of 1/3?
2. What would be the new capacity if 600 ml were added instead of 500 ml?
3. How can you verify the solution by substituting the value of $C$ back into the problem?
4. How does the fraction of the jar filled change if we add 1000 ml instead of 500 ml?
5. What would happen if the jar were 3/4 full after adding 500 ml?

Tip: When dealing with fractions in equations, multiplying through by the least common multiple can simplify the problem by eliminating the fractions.