Let's denote the total capacity of the jar as $C$ 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 $\frac{5}{6}$ full, meaning it contains $\frac{5C}{6}$ milliliters of juice.

We can set up the equation:

$\frac{C}{3} + 500 = \frac{5C}{6}$

To solve for $C$, first eliminate the fractions by multiplying the entire equation by the least common multiple of 3 and 6, which is 6:

$6 \times \left(\frac{C}{3}\right) + 6 \times 500 = 6 \times \left(\frac{5C}{6}\right)$

This simplifies to:

$2C + 3000 = 5C$

Next, subtract $2C$ from both sides:

$3000 = 3C$

Finally, solve for $C$:

$C = \frac{3000}{3} = 1000 \, \text{milliliters}$

So, the total capacity of the jar is 1000 milliliters.

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

Here are 5 related questions:

1. How would the capacity change if the jar were 2/3 full after adding 500 ml instead of 5/6?
2. What if the jar were initially 1/4 full instead of 1/3?
3. How can you verify the solution by checking the initial and final volumes?
4. How much juice would be needed to fill the jar completely from its initial state?
5. What would be the capacity if adding 1000 ml made the jar completely full?

Tip: When solving equations involving fractions, always check that your steps logically connect by considering the physical meaning (e.g., what the jar's fractions represent).