a woman is four times older than her daughter five years ago the product of their age was 175 find their present age
W = 175/D
W-5 = 4(D-5)
Substitute 175/D for W in the second equation and solve for W. Insert that value into the first equation to solve for D. Check by putting both values into the second equation.
To find their present age, let's assign variables to the unknowns:
Let's say the daughter's present age is represented by "D".
And the woman's present age is represented by "W".
From the given information, we can derive two equations:
1) "A woman is four times older than her daughter."
This can be written as: W = 4D.
2) "Five years ago, the product of their ages was 175."
This means that 5 years ago, (W - 5) * (D - 5) = 175.
To solve this system of equations, we can substitute the value of W from equation 1 into equation 2:
(4D - 5) * (D - 5) = 175.
Expanding the equation:
4D^2 - 20D - 5D + 25 = 175.
Combining like terms:
4D^2 - 25D + 25 = 175.
Rearranging the equation:
4D^2 - 25D -150 = 0.
Now, let's apply the quadratic formula to solve for D:
D = (-b ± √(b^2 - 4ac)) / 2a.
In this case:
a = 4, b = -25, and c = -150.
Plugging in the values:
D = (-(-25) ± √((-25)^2 - 4(4)(-150))) / (2 * 4).
D = (25 ± √(625 + 2400)) / 8.
D = (25 ± √3025) / 8.
D = (25 ± 55) / 8.
So, we have two possible results for D:
1) D = (25 + 55) / 8 = 80 / 8 = 10.
2) D = (25 - 55) / 8 = -30 / 8 = -3.75.
Since age cannot be negative, we discard the second solution.
Therefore, the daughter's present age (D) is 10.
Now, using equation 1, we can find the woman's present age (W):
W = 4D = 4 * 10 = 40.
So, the woman's present age is 40 and the daughter's present age is 10.