the ages of two children are 11 and
8 years. in how many years time will the product of their ages be 208
(11+x)(8+x) = 208
88 + 19x + x^2 = 208
x^2 +19x - 120 = 0
Factor and solve.
5 YEARS
Let's assume x represents the number of years in which the product of their ages will be 208.
After x years, the age of the first child will be 11 + x, and the age of the second child will be 8 + x.
The product of their ages after x years can be written as (11 + x)(8 + x).
According to the given information, we can set up the equation:
(11 + x)(8 + x) = 208
Expanding the equation, we get:
88 + 19x + x^2 = 208
Rearranging the equation, we have:
x^2 + 19x + 88 = 208
Subtracting 208 from both sides of the equation, we get:
x^2 + 19x - 120 = 0
Now we need to factorize this quadratic equation.
The factors that multiply to -120 and add up to +19 are:
(-8) and (+15)
So, we can rewrite the equation as:
(x - 8)(x + 15) = 0
This equation can be satisfied when:
x - 8 = 0 or x + 15 = 0
Solving for x in both cases, we have:
x = 8 or x = -15
Since time cannot be negative, we discard the negative value of x and conclude that in 8 years' time, the product of their ages will be 208.
To find out how many years it will take for the product of their ages to be 208, we can set up an equation.
Let's represent the current ages of the two children as x and y.
Given that one child is 11 years old and the other is 8 years old, we can write the following equations:
x = 11 (Age of one child)
y = 8 (Age of the other child)
To find out how many years it will take for the product of their ages to be 208, we add a variable for the number of years, represented by t.
After t years, the ages of the two children will be x + t and y + t.
The product of their ages after t years can be represented as:
(x + t) * (y + t) = 208
Now we can solve this equation to find the value of t.
Expanding the equation gives us:
xy + xt + yt + t^2 = 208
Rearranging the equation, we get:
t^2 + (x + y)t + xy - 208 = 0
Substituting the values of x and y, we get:
t^2 + (11 + 8)t + (11 * 8) - 208 = 0
Simplifying the equation further, we get:
t^2 + 19t + 88 - 208 = 0
Combining like terms, we have:
t^2 + 19t - 120 = 0
Now we can factorize or use the quadratic formula to solve this equation for t. By factoring, we can rewrite:
(t + 15)(t - 8) = 0
This means that either (t + 15) = 0 or (t - 8) = 0.
Solving these two equations separately, we find two possible values for t:
t + 15 = 0 --> t = -15
t - 8 = 0 --> t = 8
Since we're looking for a positive number of years, we can discard the negative value. Therefore, it will take 8 years for the product of their ages to be 208.