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
still 12
To solve this problem, let's denote the son's age as x.
According to the given information, the man is five times older than his son. Therefore, the man's age can be expressed as 5x.
Now, let's set up an equation using the information that the product of their ages four years ago was 448.
(5x - 4) * (x - 4) = 448
Expanding the equation:
5x^2 - 24x + 16 = 448
Rearranging the equation:
5x^2 - 24x - 432 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula to find the value of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 5, b = -24, and c = -432.
x = (-(-24) ± √((-24)^2 - 4 * 5 * -432)) / (2 * 5)
x = (24 ± √(576 + 8640)) / 10
x = (24 ± √(9216)) / 10
x = (24 ± 96) / 10
x = (24 + 96) / 10 or x = (24 - 96) / 10
x = 120 / 10 or x = -72 / 10
x = 12 or x = -7.2
Since the age cannot be negative, we take x = 12 as the son's age.
Now, let's find the man's age:
Man's age = 5x = 5 * 12 = 60
Therefore, the son's present age is 12, and the man's present age is 60.
To solve this problem, we can set up an equation based on the given information.
Let's assume the present age of the son is x. Therefore, the present age of the man would be 5x, as mentioned in the question.
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 can write the equation as:
(x - 4)(5x - 4) = 448
To find the present ages, we need to solve this equation.
Expanding the equation, we get:
5x^2 - 24x + 16 = 448
Rearranging the equation and simplifying, we get:
5x^2 - 24x - 432 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. Since factoring might not be straightforward, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 5, b = -24, and c = -432.
Substituting the values into 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, we get two possible values for x:
x₁ = (24 + 96) / 10 = 120 / 10 = 12
x₂ = (24 - 96) / 10 = -72 / 10 = -7.2
Since the age cannot be negative, we can discard the second solution and conclude that the present age of the son (x) is 12.
Substituting the value of x back into the equation for the man's age (5x), we get:
Man's age = 5 * 12 = 60
Therefore, the present ages are:
Son's age = 12
Man's age = 60