杰米和伊恩有一些弹珠。杰米比伊恩少 120 个弹珠。在伊恩把自己的 $\frac{1}{3}$ 弹珠给杰米后，杰米比伊恩多 24 个弹珠。伊恩最开始有多少个弹珠？

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 = $3 \times 72 = 216$.