in 8 years rachel will be twice as old as warren. Today the product of their ages is 384. how old is rachel and how old is warred?
R + 8 = 2(W + 8)= 2W + 16
R*W = 384
Solve those two equations in two unknowns.
R = 2W + 8
(2W + 8)*W = 384
2W^2 + 8W -384 = 0
W^2 + 4W -192 = 0
(W + 16)(W - 12)= 0
W = 12 is the positive-value solution. Ignore the negative solution.
R = 32
To solve this problem, let's break it down into steps:
Step 1: Set up equations
Let's assign variables to represent Rachel's and Warren's ages. Let's say Rachel's age is represented by 'R' and Warren's age is represented by 'W'.
According to the problem, in 8 years, Rachel will be twice as old as Warren. We can express this as:
R + 8 = 2(W + 8)
Also, we are given that the product of their ages today is 384. We can express this as:
R * W = 384
Step 2: Solve the equations
Let's simplify the equation R + 8 = 2(W + 8):
R + 8 = 2W + 16
R = 2W + 8 - 16
R = 2W - 8
Now, let's substitute this value of R in the equation R * W = 384:
(2W - 8) * W = 384
2W^2 - 8W = 384
2W^2 - 8W - 384 = 0
To solve this quadratic equation, we can factorize it or use the quadratic formula. Factoring might be a bit complex in this case, so let's use the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 2, b = -8, and c = -384. Plugging these values into the formula:
W = (-(-8) ± √((-8)^2 - 4 * 2 * (-384))) / (2 * 2)
Simplifying:
W = (8 ± √(64 + 3072)) / 4
W = (8 ± √3136) / 4
W = (8 ± 56) / 4
We have two possible solutions:
W = (8 + 56) / 4 = 64 / 4 = 16
W = (8 - 56) / 4 = -48 / 4 = -12
Since age cannot be negative, we discard the solution W = -12.
So, Warren is 16 years old.
Step 3: Find Rachel's age
Now that we know Warren's age, we can substitute this value back into the equation R = 2W - 8:
R = 2 * 16 - 8
R = 32 - 8
R = 24
Thus, Rachel is 24 years old.
Therefore, Rachel is 24 years old and Warren is 16 years old.