Peter, Paul and Mary have $24 between them. Together Peter and Mary have double the amount of money that Paul has, while Paul and Mary together have the same amount of money as Peter. How much money does Peter have?
21
Let x,y,z be the amounts of the three, in the order named. Then
x+y+z = 24
x+z = 2y
y+z = x
Now solve for x.
Hint: it is not 21
x+y+z=24
-x+y+z=0
2y+2z=24
y+z=12
x=12
To find out how much money Peter has, let's break down the information given in the problem step by step.
Let's assume that Peter has x dollars, Paul has y dollars, and Mary has z dollars.
1. The problem states that Peter, Paul, and Mary have $24 in total, so the first equation we can write is:
x + y + z = 24
2. It is also given that Peter and Mary have double the amount of money that Paul has. So we can write the second equation as:
x + z = 2y
3. The problem further states that Paul and Mary together have the same amount of money as Peter, which gives us the third equation:
y + z = x
Now we have a system of three equations, and we can solve it using algebraic methods. We will use the substitution method to find the value of x (the amount of money Peter has).
From equation 3, we can write:
y = x - z
Now substitute this value of y in equation 2:
x + z = 2(x - z)
Expand the equation:
x + z = 2x - 2z
Rearrange the terms:
3z = x
Substitute this value of x in equation 1:
(3z) + y + z = 24
Combine like terms:
4z + y = 24
We know that Peter, Paul, and Mary have $24 in total, so y + z + x = 24. Therefore, we can rewrite equation 4 as:
4z + (x - z) = 24
Simplify:
3z + x = 24
Substitute the value of x from equation 3:
3z + (y + z) = 24
Combine like terms:
4z + y = 24
This equation matches equation 4, so we have a system of two equations:
4z + y = 24 (equation 4)
4z + y = 24 (equation 5)
From equations 4 and 5, we can conclude that Peter has 24 dollars.
Therefore, Peter has $24.