Davis had $240 more than Ethan. Davis gave 75% of his money to Ethan
Ethan then gave 1/3 of his money to Davis. In the end, Ethan had $216 more than Davis. How much money did both of them have altogether?
Let x be the amount of money Ethan had initially. Then Davis had x/4 dollars initially. Ethan gave 1/3 of his money to Davis, so he gave (1/3)x dollars. After this, Ethan had (2/3)x dollars left, while Davis had (x/4 + 1/3)x dollars. We know that Ethan had $216 more than Davis in the end, so:
(2/3)x = (x/4 + 1/3)x + 216
Multiplying both sides by 12 gives:
8x = 3x + 2592
Solving for x gives:
x = $648
Therefore, Ethan had (2/3)($648) = $432 left, while Davis had (1/3)($648) + ($648/4) = $216 + $162 = $378. Together, they had $810 altogether.
Answer: $810
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Let's start by setting up some equations.
Let's say Davis had x dollars, and Ethan had x - $240 dollars.
Davis gave 75% of his money to Ethan, which means he gave 0.75x dollars to Ethan.
After that, Ethan had x - $240 + 0.75x = 1.75x - $240 dollars.
Ethan then gave 1/3 of his money to Davis, which means he gave (1/3)(1.75x - $240) = 0.5833x - $80 to Davis.
After that, Davis had 0.75x + 0.5833x - $80 = 1.3333x - $80 dollars.
And Ethan had 1.75x - $240 - (0.5833x - $80) = 1.1667x - $160 dollars.
We know that Ethan had $216 more than Davis in the end, so:
1.1667x - $160 = 1.3333x - $80 + $216
Simplifying this equation:
$136 = 0.1666x + $296
Subtracting $296 from both sides:
-$160 = 0.1666x
Dividing both sides by 0.1666:
x = -$960
Uh oh - this doesn't make sense! We can't have a negative amount of money.
So let's backtrack and see where we went wrong.
Our initial equation was x - $240 = Ethan's amount of money. That means Ethan had less money than Davis, which makes sense since Davis gave him 75% of his money.
So let's switch our equations to make that work:
Let's say Ethan had x dollars, and Davis had x + $240 dollars.
Davis gave 75% of his money to Ethan, which means he gave 0.75(x + $240) dollars to Ethan.
After that, Ethan had x + 0.75x + $180 = 1.75x + $180 dollars.
Ethan then gave 1/3 of his money to Davis, which means he gave (1/3)(1.75x + $180) = 0.5833x + $60 to Davis.
After that, Davis had 0.75(x + $240) + 0.5833x + $60 = 1.3333x + $270 dollars.
And Ethan had 1.75x + $180 - (0.5833x + $60) = 1.1667x + $120 dollars.
We know that Ethan had $216 more than Davis in the end, so:
1.1667x + $120 = 1.3333x + $270 + $216
Simplifying this equation:
-$366 = 0.1666x
Dividing both sides by 0.1666:
x = -$2196
Again, that's a negative amount of money. But this time we can see where we went wrong - those fractions made the numbers messy and hard to work with.
So let's try a different approach:
Let's say Davis had y dollars, and Ethan had z dollars. We don't know the exact amounts yet, but we know that:
- y = z + $240 (since Davis had $240 more than Ethan)
- Ethan got 75% of Davis's money, so Davis gave him 0.75y dollars and kept 0.25y dollars
- After that, Ethan had z + 0.75y dollars and Davis had 0.25y dollars
- Ethan gave 1/3 of his money to Davis, so he gave (1/3)(z + 0.75y) dollars to Davis
- After that, Davis had 0.25y + (1/3)(z + 0.75y) dollars = 0.25y + 0.3333z + 0.25y = 0.5y + 0.3333z dollars
- Ethan had z - (1/3)(z + 0.75y) dollars = 0.6667z - 0.25y dollars
- We know that Ethan had $216 more than Davis in the end, so:
0.6667z - 0.25y = y + $216
Now we have two equations and two variables, so we can solve for y and z:
- y = z + $240 (from the first bullet point)
- 0.6667z - 0.25y = y + $216
Let's simplify that second equation by multiplying everything by 4:
2.6668z - y = 4y + $864
Adding y to both sides:
2.6668z = 5y + $864
Substituting y = z + $240:
2.6668z = 5(z + $240) + $864
Simplifying:
2.6668z = 5z + $2160
Subtracting 2.6668z from both sides:
2.3332z = $2160
Dividing both sides by 2.3332:
z = $926.09
Substituting that back into y = z + $240:
y = $926.09 + $240 = $1166.09
So Davis had $1166.09 and Ethan had $926.09.
Altogether, they had:
$1166.09 + $926.09 = $2092.18
So both of them had $2092.18 altogether.
Let's break down the given information step by step to find the total amount of money they had altogether.
1) Davis had $240 more than Ethan:
Let's assume Ethan had x dollars. Therefore, Davis had x + $240 dollars.
2) Davis gave 75% of his money to Ethan:
Davis gave away (75/100) * (x + $240) = (3/4) * (x + $240) dollars to Ethan.
After this transaction, Davis had (1-3/4) * (x + $240) = (1/4) * (x + $240) dollars left.
3) Ethan then gave 1/3 of his money to Davis:
Ethan gave away (1/3) * x dollars to Davis.
After this transaction, Ethan had (1-1/3) * x = (2/3) * x dollars left.
4) In the end, Ethan had $216 more than Davis:
According to the given information, Ethan had $216 more than Davis.
This implies that (2/3) * x - (1/4) * (x + $240) = $216.
Now, let's solve this equation to find the value of x, which represents the amount of money Ethan had initially.