the sum of the ages of three redwood trees is exactly one thousand years. when the youngest tree reaches the age of the middle tree, the middle tree will have reached the age of the oldest tree. at that time the middle tree will be four times the current age of the youngest tree. determine the current age of each tree.
younger tree --- x
middle tree -----y
older tree -------z
x+y+z = 1000 *
time for younger tree to reach age of current middle tree = y - x
at that point:
younger tree --- y
middle tree --- y + y-x = 2y - x
older tree ---- z + y-x
2y-x = z
2y-x = 4x
2y = 5x
y = 5x/2
x + y + z = 1000
x + (5x/2) + (2y - x) = 1000
x + 5x/2 + 10x/2 - x = 1000
15x/2 = 1000
15x = 2000
x = 2000/15 years = 400/3 = appr 133.3 years
y = 5000/15 years = 1000/3 = appr 333.3 years
z = 8000/15 = 1600/3 years = appr 533.3 years
check:
in 200 years the younger will be as old as the middle is now
then the middle would be 533.3, which is the same as the older is now, check!
and which is 4 times the 133.3
Let's denote the ages of the three redwood trees as follows:
Youngest tree = x years
Middle tree = y years
Oldest tree = z years
We have been given a few conditions:
1. The sum of the ages of the three trees is exactly one thousand years: x + y + z = 1000.
2. When the youngest tree reaches the age of the middle tree, the middle tree will have reached the age of the oldest tree: x + y = z.
3. At that time, the middle tree will be four times the current age of the youngest tree: y = 4x.
Now, we can solve these equations simultaneously to find the ages of the trees.
From equation 3, we can express y in terms of x: y = 4x.
Substituting this value of y into equation 2, we get: x + 4x = z.
Combining like terms, we have: 5x = z. (Equation 4)
We can substitute this value of z back into equation 1 to eliminate z:
x + y + z = 1000
x + 4x + 5x = 1000
Combining like terms, we have: 10x = 1000.
Dividing both sides by 10, we get: x = 100.
Now, we can substitute this value of x into equation 4 to find the value of z:
5x = z
5 * 100 = z
500 = z
So, the current age of the oldest tree (z) is 500 years.
Now, we can substitute the value of x back into equation 3 to find the value of y:
y = 4x
y = 4 * 100
y = 400
Thus, the current age of the middle tree (y) is 400 years.
Therefore, the current ages of the three redwood trees are:
Youngest tree: 100 years
Middle tree: 400 years
Oldest tree: 500 years
To solve this problem, let's assume the current ages of the three redwood trees are:
Youngest tree: x years old
Middle tree: y years old
Oldest tree: z years old
According to the given information, the sum of the ages of the three trees is 1000 years:
x + y + z = 1000 --(1)
We also know that when the youngest tree (x) reaches the age of the middle tree (y), the middle tree will have reached the age of the oldest tree (z):
x + y = z --(2)
Furthermore, at that time (when the youngest tree reaches the age of the middle tree), the middle tree will be four times the current age of the youngest tree:
y = 4x --(3)
Now, we have a system of three equations (1), (2), and (3), which we can solve simultaneously.
To solve this system of equations, we can start by substituting the value of y from equation (3) into equations (1) and (2).
From equation (3), we have:
y = 4x
Substituting y = 4x into equation (1):
x + 4x + z = 1000
Simplifying, we get:
5x + z = 1000 --(4)
Substituting y = 4x into equation (2):
x + 4x = z
Simplifying, we get:
5x = z --(5)
Now, we have two equations (4) and (5) with two variables (x and z). We can solve for their values.
Let's rearrange equation (5) to get z in terms of x:
z = 5x
Substituting this into equation (4):
5x + 5x = 1000
Combining like terms, we get:
10x = 1000
Dividing both sides by 10, we find:
x = 100
Now that we have the value of x, we can find the values of y and z.
From equation (3):
y = 4x
y = 4 * 100
y = 400
From equation (5):
z = 5x
z = 5 * 100
z = 500
Therefore, the current ages of the three redwood trees are:
Youngest tree: 100 years old
Middle tree: 400 years old
Oldest tree: 500 years old