Dorthy is 6 years older than Ricardo. The product of their present age is twice what the product of their ages was 6 years ago. How old is Dorthy?
Ricardo's present age --- x
Dorothy's present age --- x+6
product of their present ages = x(x+6)
6 years ago:
Ricardo x-6
Dorothy x
product of their ages then = x(x-6)
x(x+6) = 2x(x-6)
solve for x, sub into x+6
Let D=age of Dorthy.
" Dorthy is 6 years older than Ricardo. "
Age of Ricardo = D-6
"The product of their present age"
D(D-6)
"is twice what the product of their ages was 6 years ago"
=2*((D-6)*(D-6-6))
Thus
D(D-6) = 2*((D-6)*(D-12))
Simplify and solve for D and reject the solution of D=6 because it is a trivial solution. Check the answer by substitution into problem data.
To solve this problem, we can use algebraic equations. Let's assign variables to the ages of Dorthy and Ricardo.
Let's assume Dorthy's age is represented by the variable "D," and Ricardo's age is represented by the variable "R."
We are given two pieces of information:
1. "Dorthy is 6 years older than Ricardo." This can be written as: D = R + 6.
2. "The product of their present age is twice what the product of their ages was 6 years ago." This can be written as: D * R = 2 * (D - 6) * (R - 6).
Now, we can solve these equations simultaneously to find the values of D and R.
Substituting the value of D from equation 1 into equation 2, we get:
(R + 6) * R = 2 * ((R + 6) - 6) * (R - 6)
Simplifying further:
R^2 + 6R = 2 * R * (R - 6)
Expanding and simplifying the equation:
R^2 + 6R = 2R^2 - 12R
Rearranging and simplifying again:
R^2 - 18R = 0
Factoring out R:
R(R - 18) = 0
This equation has two solutions: R = 0 and R = 18.
Since age cannot be zero, we discard R = 0. Therefore, R = 18.
Now, substituting this value of R into equation 1 to find D:
D = R + 6
D = 18 + 6
D = 24
Therefore, Dorthy is 24 years old.