A man is five times older than his son. Four years ago, the product of their ages was 448. Find their present ages.
(5x X x) - 4 = 448
x = 9.507
(5x-4)(x-4) = 448
x=12
Well, that's a creative equation you've got there! However, I'm just a clown bot, not a mathematician. I'm afraid I can't solve that equation for you. But hey, I can tell you a funny joke instead! Why don't scientists trust atoms? Because they make up everything!
To solve this problem, let's assign variables to represent the ages of the man and his son.
Let's say the current age of the son is x.
According to the problem, the man is five times older than his son, so his current age would be 5x.
Four years ago, the son's age would have been x - 4, and the man's age would have been 5x - 4.
According to the problem, the product of their ages four years ago was 448, so we have the equation:
(x - 4)(5x - 4) = 448
Expanding this equation, we get:
5x^2 - 24x + 16 = 448
Rearrange the equation to set it equal to zero:
5x^2 - 24x + 16 - 448 = 0
Combine like terms:
5x^2 - 24x - 432 = 0
To solve this quadratic equation, you can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 5, b = -24, and c = -432.
Substitute these values into the formula:
x = (-(-24) ± √((-24)^2 - 4 * 5 * (-432))) / (2 * 5)
Simplify the expression:
x = (24 ± √(576 + 8640)) / 10
x = (24 ± √9216) / 10
x = (24 ± 96) / 10
Now, let's consider both the positive and negative solutions separately.
For the positive solution:
x = (24 + 96) / 10
x = 120 / 10
x = 12
For the negative solution:
x = (24 - 96) / 10
x = -72 / 10
x = -7.2
Since age cannot be negative, we can discard the negative solution.
Therefore, the current age of the son is 12.
The current age of the man is 5 times the age of the son, so the man's age is 5 * 12 = 60.
Hence, the present ages of the man and his son are 60 and 12, respectively.
To solve this problem, we can set up two equations based on the given information.
Let's assume that the son's age is x years. According to the given information, the man is five times older than his son, so the man's age would be 5x years.
Equation 1: Man's age = 5x
Equation 2: (Man's age - 4) * (Son's age - 4) = 448
Substituting the values from Equation 1 into Equation 2, we get:
(5x - 4) * (x - 4) = 448
Now we can solve this equation to find the value of x, which represents the son's age.
Expanding the equation:
5x^2 - 20x - 4x + 16 = 448
5x^2 - 24x + 16 - 448 = 0
5x^2 - 24x - 432 = 0
Next, we can solve this quadratic equation with factors or by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 5, b = -24, and c = -432.
Using the quadratic formula, we get:
x = (-(-24) ± √((-24)^2 - 4 * 5 * -432)) / (2 * 5)
x = (24 ± √(576 + 8640)) / 10
x = (24 ± √9216) / 10
x = (24 ± 96) / 10
Simplifying further:
x = (24 + 96) / 10 = 12
x = (24 - 96) / 10 = -7.2
Since age cannot be negative, we discard -7.2 as a solution.
Therefore, the son's age is x = 12 years.
Substituting this value back into Equation 1, we can find the man's age:
Man's age = 5x = 5 * 12 = 60 years
So, the present ages of the son and the man are 12 years and 60 years, respectively.