The sum of ages of two sisters are 19 years.in five years time the product of their ages will be 208.what are their ages?
older sister --- x
younger sister --- 19-x
in 5 years:
(x+5)(24-x) = 208
-x^2 + 19x + 120 - 208 = 0
x^2 - 19x + 88 = 0
(x-11)(x-8) = 0
x = 11 or x = 8
older sis is 11, younger is 8
check:
in 5 years:
16(13) = 208 , all is good
x+y = 19
(x+5)(y+5) = 208
now just solve for x and y.
11 and 8
Correct
To find the ages of the two sisters, let's use algebraic equations:
Let's assume the age of the younger sister is denoted as "x" and the age of the older sister is denoted as "y".
According to the given information, the sum of their ages is 19. Therefore, we can write the following equation: x + y = 19.
In five years, their ages will increase by 5. So, we can write the following equation for the product of their ages in 5 years: (x + 5)(y + 5) = 208.
Now, we can use these two equations to solve for the ages of the sisters:
From the first equation, we can rewrite it as x = 19 - y and substitute it into the second equation:
(19 - y + 5)(y + 5) = 208
(24 - y)(y + 5) = 208
240 - 24y + 5y - y^2 = 208
240 - 19y - y^2 = 208
0 = y^2 - 19y - 32
Now, we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring this equation does not yield integer solutions, so we'll use the quadratic formula:
y = [-(-19) ± √((-19)^2 - 4(1)(-32))] / (2*1)
y = [19 ± √(361 + 128)] / 2
y = [19 ± √489] / 2
Now, we need to check which value of y gives us an integer solution for x. The possible values for y are:
y = [19 + √489] / 2 ≈ 15.35
y = [19 - √489] / 2 ≈ 3.65
The second value, y ≈ 3.65, gives us an integer solution for x. We need to check the corresponding value of x:
x = 19 - y
x = 19 - 3.65 ≈ 15.35
Therefore, the ages of the two sisters are approximately 15 and 4, respectively.