Mary is now three times as old as her daughter. Four years ago the products of their age was 40 .calculate their present years
now:
daughter ---- x years
Mary -------- 3x years
4 years ago:
daughter : x-4
Mary : 3x-4
(x-4)(3x-4) = 40
3x^2 -16x + 16 = 40
3x^2 - 16x - 24 = 0
I was expecting it to factor, hoping for whole number solution, but ....
x = (16 ± √544)/6
= 6.55 or a negative
check:
if the daughter is appr 6.55
then Mary is 19.65 years old
Four years ago, Mary was 15.65 and the daughter was 2.55 , and 15.65(2.55) = 39.9 or
appr 40
The math works out, but Mary was 13 when the daughter was born. Somebody should be in jail.
To solve this problem, we can start by assigning variables to represent the present ages of Mary and her daughter. Let's say Mary's age is represented by M, and her daughter's age is represented by D.
According to the problem, Mary is now three times as old as her daughter. This can be written as an equation: M = 3D.
The problem also states that four years ago, the product of their ages was 40. To represent this, we need to subtract 4 from their ages and multiply them together: (M-4)(D-4) = 40.
Now we have two equations:
1. M = 3D
2. (M-4)(D-4) = 40
We can solve this system of equations using substitution or elimination.
Using substitution, we can substitute the value of M from the first equation into the second equation:
(3D-4)(D-4) = 40
Expanding the equation:
3D^2 - 12D - 4D + 16 = 40
Combining like terms:
3D^2 - 16D + 16 = 40
Rearranging the equation:
3D^2 - 16D - 24 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
Factoring the quadratic equation is not possible in this case, so we'll use the quadratic formula:
D = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -16, and c = -24. Plugging in these values:
D = (-(-16) ± √((-16)^2 - 4*3*(-24))) / (2*3)
Simplifying:
D = (16 ± √(256 + 288)) / 6
D = (16 ± √544) / 6
We then find the two possible values of D:
D ≈ (16 + √544) / 6
D ≈ (16 - √544) / 6
Calculating these values gives us:
D ≈ 9
D ≈ -2.5
Since ages cannot be negative, we disregard D ≈ -2.5 and only consider D ≈ 9.
Now we can substitute this value of D back into one of the original equations to find M. Let's use the equation M = 3D:
M = 3 * 9
M = 27
Therefore, the present age of Mary is 27 and the present age of her daughter is 9.