Ali, Brandon, Carrie and Devi shared a sum of money equally at first. Ali gave $\frac{2}{3}$ of his money to Brandon.

1. Determine Ali's initial fraction of the total sum

Since the money was shared equally among 4 people, Ali initially had $\frac{1}{4}$ of the total sum.

2. Calculate the fraction of his own money Ali had left

Ali gave away $\frac{2}{3}$ of his money, so he had $1 - \frac{2}{3} = \frac{1}{3}$ of his money left.

3. Calculate the fraction of the total sum Ali had in the end

Fraction of the total sum = $\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$

1. Establish initial units for each person

Let the initial amount each person had be $3$ units (to easily divide by $3$).
Total sum = $12$ units.
Ali: $3$ units
Brandon: $3$ units
Carrie: $3$ units
Devi: $3$ units

2. Track Ali's transfer to Brandon

Ali gives $\frac{2}{3}$ of his money to Brandon.
Ali gives $\frac{2}{3} \times 3 = 2$ units.
Brandon now has $3 + 2 = 5$ units.

3. Track Brandon's transfer to Carrie

Brandon gives $\frac{1}{5}$ of his money to Carrie.
Brandon gives $\frac{1}{5} \times 5 = 1$ unit.
Carrie now has $3 + 1 = 4$ units.

4. Track Carrie's transfer to Devi and set up equation for Devi

Carrie gives $\frac{3}{8}$ of her money and an additional $\char36{}55$ to Devi.
Carrie gives $\frac{3}{8} \times 4 = 1.5$ units plus $\char36{}55$.
Devi receives $1.5$ units $+ \char36{}55$.
Devi's final amount = Initial $3$ units $+ 1.5$ units $+ \char36{}55 = 4.5$ units $+ \char36{}55$.

5. Solve for 1 unit and find the total sum

Devi had $\char36{}595$ in the end.
$4.5$ units $+ \char36{}55 = \char36{}595$
$4.5$ units $= \char36{}595 - \char36{}55 = \char36{}540$
$1$ unit $= \char36{}540 \div 4.5 = \char36{}120$
Total sum $= 12$ units $= 12 \times \char36{}120 = \char36{}1440$