Jamie and Ian had some marbles. Jamie had 120 fewer marbles than Ian. After Ian gave $\frac{1}{3}$ of his marbles to Jamie, Jamie had 24 more marbles than Ian. How many marbles did Ian have at first?

1. Represent the initial number of marbles using units

Let Ian's initial number of marbles be $3$ units (since he gives away $\frac{1}{3}$ of them).
Jamie's initial number of marbles = $3$ units $- 120$.

2. Determine the number of units each person has after the transfer

Ian gives $\frac{1}{3}$ of his marbles, which is $1$ unit, to Jamie.
Ian's final marbles = $3$ units $- 1$ unit = $2$ units.
Jamie's final marbles = ($3$ units $- 120$) $+ 1$ unit = $4$ units $- 120$.

3. Use the final difference to find the value of 1 unit

Jamie had $24$ more marbles than Ian in the end:
($4$ units $- 120$) $- 2$ units = $24$
$2$ units $- 120 = 24$
$2$ units = $144$
$1$ unit = $72$

4. Calculate Ian's initial number of marbles

Ian's initial marbles = $3$ units = $3 \times 72 = 216$.