A father was three times as old as his son fifteen years ago and four times his sons age nineteen years ago. When was the father twice as old as the son?
f-15 = 3(s-15)
f-19 = 4(s-19)
solve for s and f, and then find x such that
f-x = 2(s-x)
To solve the problem, let's represent the current age of the son as x and the current age of the father as y.
According to the problem, "A father was three times as old as his son fifteen years ago." This can be expressed as y - 15 = 3(x - 15).
The second piece of information states that "the father was four times his son's age nineteen years ago." This can be expressed as y - 19 = 4(x - 19).
We can now solve the two equations to find the current ages of the son and father.
We first simplify the first equation: y - 15 = 3x - 45.
Next, we simplify the second equation: y - 19 = 4x - 76.
Rearranging each equation, we have:
y - 3x = -30 (Equation 1)
y - 4x = -57 (Equation 2)
We can solve this system of linear equations by substitution or elimination. Let's use elimination:
Multiply Equation 1 by 4: 4(y - 3x) = 4(-30), which simplifies to 4y - 12x = -120.
Multiply Equation 2 by 3: 3(y - 4x) = 3(-57), which simplifies to 3y - 12x = -171.
Subtract Equation 1 from Equation 2: (3y - 12x) - (4y - 12x) = -171 - (-120),
which simplifies to -y = -51.
Divide both sides of the equation by -1: y = 51.
Now substitute the value of y (51) into Equation 1: 51 - 3x = -30.
Solve for x: -3x = -30 - 51, which simplifies to -3x = -81.
Divide both sides by -3: x = 27.
Therefore, the current age of the son (x) is 27 years old and the current age of the father (y) is 51 years old.
To find when the father was twice as old as the son, we need to look for a time difference between their ages where the father's age is twice that of the son's.
Let's assume that t years ago, the father was twice the age of the son.
So, the father's age t years ago would be (y - t), and the son's age t years ago would be (x - t).
Therefore, we can set up the equation: y - t = 2(x - t).
Let's solve for t:
Substitute the current ages:
51 - t = 2(27 - t).
Simplify the equation:
51 - t = 54 - 2t.
Add t to both sides:
51 = 54 - t.
Subtract 54 from both sides:
-3 = -t.
Multiply by -1 to get t:
t = 3.
So, three years ago, the father was twice as old as his son.
Thus, the answer is: The father was twice as old as the son three years ago.