Tim has 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?
Let X be the amount of money Peter has.
So, Tim has 2/3*X = (2*X)/3 as much money
After Tim received $120 from his mother, he has (2*X)/3 + 120 = 2*(X - 120) as much money.
Multiplying through the parentheses, we get (2*X)/3 + 120 = 2*X -240
Multiplying through by 3, we get 2*X + 360 = 6*X - 720
Subtracting 2*X and 360, we get 0 = 4*X - 1080
Adding 1080 to both sides, we get 1080 = 4*X
Dividing both sides by 4, we get X = 270
Then Tim had (2*270)/3 = $<<2*270/3=180>>180 at the start. Answer: \boxed{180}.
Let's assume that Peter has x dollars.
According to the given information, Tim has 2/3 of Peter's money, which is (2/3)x dollars.
After Tim received $120 from his mother, his total money becomes (2/3)x + $120.
Peter used $120, so his total money becomes x - $120.
According to the problem, after these transactions, Tim had twice as much money as Peter, which can be represented as:
(2/3)x + $120 = 2(x - $120).
Now, we can solve this equation to find the value of x:
2/3x + $120 = 2x - $240.
To remove the fractions, we can multiply both sides of the equation by 3:
2x + 360 = 6x - $720.
Subtracting 2x from both sides:
360 = 4x - $720.
Adding $720 to both sides:
1080 = 4x.
Dividing both sides by 4:
x = 1080/4.
x = $270.
Therefore, Peter initially had $270.
Since Tim has 2/3 of Peter's money, Tim initially had (2/3) * $270 = $180.
So, Tim initially had $180.
To solve this problem, let's break it down step by step.
Let's assume that Peter has x dollars. Since Tim has 2/3 as much money as Peter, we can say that Tim has (2/3) * x dollars.
After Tim received $120 from his mother, his total amount of money is now (2/3) * x + $120.
On the other hand, Peter used $120, so his total amount of money is now x - $120.
According to the problem statement, Tim had twice as much money as Peter after these transactions. So, we have the following equation:
(2/3) * x + $120 = 2 * (x - $120)
Let's solve this equation for x:
2/3 * x + $120 = 2x - $240
Multiply through by 3:
2x + $360 = 6x - $720
Rearrange the equation by moving terms to one side:
4x = $1080
x = $270
Therefore, Peter initially had $270.
Since Tim has 2/3 as much money as Peter, we can calculate Tim's initial amount:
Tim's initial amount = (2/3) * $270
= (2/3) * $270
= $180
Therefore, Tim initially had $180.