The ratio of the amount of money Karen had to the amount of money Peter had was 2:5. Peter received $75 from his mother, the ratio became 4:15. How much money did each person have at first?
Karen had 2x
Peter had 5x
after mom's gift
Karen --- 2x
Peter ---- 5x+75
2x/(5x+75) = 4/15
30x = 20x + 300
10x = 300
x = 30
Karen had 60
peter had 150
check:
Karen still has 60 , Peter has 225
60:225 = 4:15
Let's assume that Karen had $2x and Peter had $5x initially, where x is the common ratio.
After Peter received $75, the ratio became 4:15, so we can set up the equation:
(2x + 75)/(5x) = 4/15
Cross multiplying, we get:
15(2x + 75) = 4(5x)
Expanding, we have:
30x + 1125 = 20x
Subtracting 20x from both sides, we get:
10x + 1125 = 0
Subtracting 1125 from both sides, we have:
10x = -1125
Dividing both sides by 10, we get:
x = -112.5
Since it doesn't make sense to have negative money, we can conclude that the initial assumption was incorrect.
Therefore, we need to reverse the calculation. Let's assume Karen had $2y and Peter had $5y initially, where y is the common ratio.
After Peter received $75, the ratio became 4:15, so we can set up the equation again:
(2y)/(5y + 75) = 4/15
Cross multiplying, we get:
15(2y) = 4(5y + 75)
Simplifying, we have:
30y = 20y + 300
Subtracting 20y from both sides, we get:
10y = 300
Dividing both sides by 10, we get:
y = 30
Now, we can substitute y back into the initial values:
Karen initially had $2y = $2(30) = $60
Peter initially had $5y = $5(30) = $150
Therefore, Karen initially had $60 and Peter initially had $150.
To solve this problem, we can set up a system of equations using the given information.
Let's assume that the initial amount of money Karen had is represented by "2x" and the initial amount of money Peter had is represented by "5x."
According to the problem, the ratio of Karen's initial money to Peter's initial money is 2:5. So, we have the equation:
Karen's initial money / Peter's initial money = 2/5
(2x) / (5x) = 2/5
Now, we know that Peter received $75 from his mother, and the ratio became 4:15. So, we have another equation:
(Karen's initial money) / (Peter's initial money + $75) = 4/15
(2x) / (5x + 75) = 4/15
Now, we can solve this system of equations to find the values of x and ultimately determine the initial amount of money for Karen and Peter.
Let's start by cross-multiplying the first equation:
2 * (5x) = (2x) * 5
10x = 10x
Since both sides of the equation are equal, it means that the first equation does not provide us with any new information. Hence, we can disregard it.
Thus, we only need to solve the second equation:
(2x) / (5x + 75) = 4/15
Cross-multiplying:
(2x) * 15 = (5x + 75) * 4
30x = 20x + 300
Simplifying the equation:
30x - 20x = 300
10x = 300
x = 30
Now that we have the value of x, we can determine the initial amount of money for Karen and Peter:
Karen's initial money = 2x = 2 * 30 = $60
Peter's initial money = 5x = 5 * 30 = $150
Therefore, Karen initially had $60, and Peter initially had $150.