Answer: The age of Ana is 37.9 years old.
Step-by-step explanation:
This can be translated to:
"Rocio tells Ana:
I have the triple of the age that you had when i had half your age.
And when you have the age that i have today, our ages will ad up to 90"
If A is the age of Ana, and R the age of Rocio, we have that:
when Rocio had half the age of Ana was k years ago
k years ago the age of Ana was A - k
R - k = A/2
R = 3*(A - k)
Ana will have the age of Rocio in h years
A + h = R
in h years, Rocio will have R + h years.
A + h + R + h = R + R + h = 90
So we have 4 equations and 4 variables.
R - k = A/2
R = 3*(A - k)
A + h = R
2R + h = 90
We can replace the third equation into the other 3.
(A + h) - k = A/2
(A + h) = 3*(A - k)
2*(A + h) + h = 90
Now we can isolate k from the first equation, and then replace it into the second equation:
A + h - k = A/2
k = A - A/2 + h = A/2 + h
and the second equation is:
(A + h) = 3*(A - k) = 3*(A - A/2 - h) = 3*(A/2 - h)
Now we have two equations:
(A + h) = 3*(A/2 - h)
2(A + h) + h = 90
now we need to isolate h in one equation and replace it into the other, let's isolate h in the second equation:
2(A + h) + h = 90
2A + 3h = 90
3h = 90 - 2A
h = 30 - (2/3)*A
now we replace it into the other equation:
(A + 30 - (2/3)*A) = 3*(A/2 - 30 + (2/3)*A)
Now we solve this equation for A.
(1/3)*A + 30 = 3*((3/6)*A - 30 + (4/6)*A) = 3*((7/6)*A - 30)
(1/3)*A + 30 = (7/2)*A - 90
A*( 7/2 - 1/3) = 30 + 90 = 120
A = 120/(7/2 - 1/3) = 37.9