SOLUTION: 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 the

Question 766411: 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?
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
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

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