Tim had $\frac{2}{3}$ as much money as Peter. After Tim received $120 from his mother and Peter used $120, Tim had twice as much money as Peter. How much money did Tim have at first? Ans: ___

1. Identify that the total amount of money remains unchanged and set up the initial and final ratios.

Since Tim receives $120 and Peter uses $120, the total amount of money they have together remains constant.
Initial ratio of Tim to Peter = $2 : 3$ (Total = $5$ units)
Final ratio of Tim to Peter = $2 : 1$ (Total = $3$ units)

2. Make the total units the same for both ratios to compare them.

LCM of $5$ and $3$ is $15$.
Multiply the initial ratio by $3$:
Initial Tim : Peter = $6 : 9$ (Total = $15$ units)
Multiply the final ratio by $5$:
Final Tim : Peter = $10 : 5$ (Total = $15$ units)

3. Find the value of 1 unit and calculate Tim's initial money.

Increase in Tim's units = $10 - 6 = 4$ units
$4 \text{ units} = \char36{}120$
$1 \text{ unit} = \char36{}120 \div 4 = \char36{}30$
Tim's initial money = $6 \text{ units} = 6 \times \char36{}30 = \char36{}180$