Robert and Damien had the same amount of money. Each day, Robert spent $4 and Damien spent $6. When Damien had $12 left, Robert had 4 times as much money as Damien. How much money did each boy have at first?
Let the amount each had at first be x
After t days
Robert had x - 4t
Damian had x - 6t
when x-6t = 12, x-4t = 4(x-6t)
1st equation ---> x = 6t+12
2nd equation:
x-4t = 4x - 24t
20t = 3x
sub in the first:
20t = 3(6t+12)
20t = 18t + 36
t = 18
then x = 6(18) + 12 = 120
They each had $ 120.00
check
after 18 days:
Robert had 120-4(18) = 48
Damien had 120 - 6(18) = 12
Does Robert have 4 times what Damien has ?? YEAHH
wow complecated
To solve this problem, let's use algebraic equations:
Let's assume that both Robert and Damien had x dollars initially.
Since Robert spends $4 each day, after d days, he would have:
Robert's remaining money = x - 4d
Similarly, Damien would have:
Damien's remaining money = x - 6d
We know that when Damien had $12 left, Robert had 4 times as much money as Damien. So we can set up the following equation:
4 * (Damien's remaining money) = Robert's remaining money - Damien's remaining money
Plugging in the values we calculated earlier, we get:
4 * (x - 6d) = (x - 4d) - (x - 6d)
Simplifying the equation:
4x - 24d = x - 4d - x + 6d
4x - 24d = 2d
4x = 26d
Since we don't have actual values for x or d, we can find the ratio of x to d by dividing both sides of the equation by d:
4x/d = 26d/d
4x/d = 26
x/d = 26/4
x/d = 6.5
This means that for every $6.50 Robert has, Damien has $1. Based on this ratio, we can conclude that Robert initially had more money than Damien.
To find the initial amount of money each boy had, we can substitute the ratio x/d = 6.5 into one of the original equations. Let's use Robert's remaining money equation (x - 4d) and solve for x:
x - 4d = 6.5d
x = 6.5d + 4d
x = 10.5d
So, initially, Robert had 10.5 times more money than Damien.
Now we need to find a value for d, which represents the number of days. Since Damien had $12 left when Robert had 4 times his money, we can set up another equation:
Robert's remaining money = Damien's remaining money + 4 * (Damien's remaining money)
Plugging in the values we have:
x - 4d = 12 + 4 * (x - 6d)
Simplifying the equation:
x - 4d = 12 + 4x - 24d
Bringing the variables to one side and numbers to the other side:
x - 4x = 12 - 24d + 4d
-3x = -20d
Dividing both sides by -20:
x/d = 3/20
Since x/d = 6.5, we can set up the following equation and solve for d:
6.5 = 3/20
20 * 6.5 = 3
130 = 3
d = 130/3
Now we can substitute the value of d back into x = 10.5d to find the initial amounts of money:
x = 10.5 * (130/3)
So, Damien initially had:
x = 10.5 * (130/3)
x = 455 dollars
And Robert initially had:
x = 10.5 * (130/3)
x = 455 dollars
Therefore, both Robert and Damien initially had $455.