Jean had some tickets to sell. After selling 56 of them in the morning and $\frac{4}{7}$ of the remainder in the afternoon, she was left with $\frac{1}{5}$ of the tickets. How many tickets were sold altogether?

1. Express the remainder and the tickets left in terms of units.

Since $\frac{4}{7}$ of the remainder was sold in the afternoon, let the remainder be $7$ units.
Tickets sold in the afternoon = $4$ units.
Tickets left = $7 - 4 = 3$ units.

2. Relate the tickets left to the total number of tickets to find the total units.

We are given that the tickets left represent $\frac{1}{5}$ of the total tickets.
$\frac{1}{5}$ of total = $3$ units.
Total tickets = $3 \times 5 = 15$ units.

3. Find the number of units sold in the morning and solve for 1 unit.

Tickets sold in the morning = Total tickets $-$ Remainder
$= 15 - 7 = 8$ units.
We know $8$ units = $56$ tickets.
$1$ unit = $56 \div 8 = 7$ tickets.

4. Calculate the total number of tickets sold altogether.

Total units sold = Total tickets $-$ Tickets left
$= 15 - 3 = 12$ units.
Total tickets sold = $12 \times 7 = 84$.