A woman is twice as old as her daughter was when she was as old as her daughter is now. When her daughter is as old as she is now, their combined ages will be 180. How old are they now?
Let W = the woman's age now
Let D = the daughter's age now
Let X = the number of years AGO
talked about in the problem
Let Y = the number of years FROM NOW
talked about in the problem.
a woman is twice as old as her daughter was X YEARS AGO
W = 2(D - X)
X YEARS AGO WAS when she (THE WOMAN) was as old as her daughter is now.
W - X = D
when her daughter is as old as she is now WHICH WILL BE Y YEARS FROM NOW
D + Y = W
Y YEARS FROM NOW, their combined ages will be 180.
(W + Y) + (D + Y) = 180
So we have this system of 4 equations
in 4 unknowns:
W = 2(D - X)
W - X = D
D + Y = W
(W + Y) + (D + Y) = 180
Simplifying each of those equations:
W - 2D + 2X = 0
W - D - X = 0
-W + D - + Y = 0
W + D + 2Y = 180
how old are they?
Solve that by elimination and get
W - 2D + 2X = 0
W - D - X = 0
-W + D - + Y = 0
W + D + 2Y = 180
Multiply the 2nd equation by 2 and add it to the
first equation to eliminate X and get
W - 2D + 2X = 0
2W - 2D - 2X = 0
-----------------------
3W - 4D = 0
Multiply the 3rd equation by -2 and add the
fourth equation to it to eliminate Y and get
2W - 2D - - 2Y = 0
W + D + 2Y = 180
-----------------------
3W - D = 180
So solve the system
3W - 4D = 0
3W - D = 180
by multiplying the first equation by -1
-3W + 4D = 0
3W - D = 180
---------------
3D = 180
D = 60
Substitute that in
3W - D = 180
3W - 60 = 180
3W = 240
W = 80
Answer: the Woman is 80 and her daughter is 60.
To check we need to find X and Y
W - X = D
80 - X = 60
-X = -20
X = 20 years ago
D + Y = W
60 + Y = 80
Y = 20 years from now
W = 80, D = 60, X = 20, Y = 20
The woman is 80, which is twice as old
as her 60-year old daughter was 20 years
ago, which is 40. And 80 is twice 40. That
checks. The woman was was 60 then, which
is as old as her daughter is now, which
indeed is 60. So that checks. 20 years
from now is when her daughter, who is now
60, will be 80 (as old as the woman is now).
That checks. The woman will then be 100,
and their combined ages will be 100+80 or
180. Everything checks.
Edwin
