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Three years ago a father was four times as old as his daughter is now. the product of their present ages 430 . calculate the present ages of the father and daughter

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Let's assume the present age of the daughter is 'd' years and the present age of the father is 'f' years.

According to the given information, three years ago, the father was four times as old as his daughter is now. So, we can set up an equation:

(f - 3) = 4d

Also, the product of their present ages is 430:

f * d = 430

Now we can solve this system of equations to find the present ages of the father and daughter.

Using substitution, we can substitute the value of (f - 3) from the first equation into the second equation:

(4d - 3) * d = 430

Expanding the equation:

4d^2 - 3d - 430 = 0

We can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. By factoring, we find:

(2d + 43)(2d - 10) = 0

Setting each factor equal to zero:

2d + 43 = 0 or 2d - 10 = 0

Solving each equation:

2d = -43 or 2d = 10

d = -43/2 or d = 10/2

Since age cannot be negative, we discard the negative solution. Therefore, the present age of the daughter (d) is 5 years.

Substituting this value back into one of the original equations, we can solve for the father's age:

f = 4d + 3 = 4 * 5 + 3 = 20 + 3 = 23 years

Hence, the present age of the father is 23 years, and the present age of the daughter is 5 years.
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