Ann is currently 44 years old and Ruth is 54. How did I get this?
R=Ruth's present age and A=Ann's present age.
There seemto be 4 time zones: y years ago, now, x years hence, z years hence. Let's start in the past. A-y=R/2. y years ago Ann was half Ruth's present age. At that time Ruth was R-y years old. Now the future. When Ann is twice as old as this in x years' time we have A+x=2(R-y). At the same time Ruth's age will be half as much again (3/2) as Ann's will be in z year's time: R+x=3/2(A+z). In z year's time Ruth will be R+z and Ann will be A+z years old, and Ruth will be half again as old as Ann is now: R+z=3/2A.
We can start to eliminate some variables. y=A-R/2 from the first equation. So A+x=2(R-A+R/2) substituting in the second equation. A+x=3R-2A. z=3/2A-R from last equation is last paragraph. So R+x=3/2(A+3/2A-R)=3/2(5/2A-R). We now have two simultaneous equations from which we can eliminate x: A-R=3R-2A-3/2(5/2A-R). A-R=9/2R-23A/4, so, getting rid of the fractions, 4A-4R=18R-23A. 27A=22R, from which A=22R/27 or R=27A/22. We know that either Ruth or Ann is aged between 50 and 59. We also know that all ages are whole numbers. The only multiple of either 22 or 27 between 50 and 59 is 54=2*27. So R=54 and A=44. Therefore Ruth is 54 and Ann is 44.
The original question can now be read knowing the women's ages to confirm their correctness. In the course of doing so, x, y and z will be established. y=17, z=12 and x=30 years. See table below where age relationships should look clearer. Every age is related to another across time. For example, Ruth's age in 12 years' time is half as much again as Ann's present age, and Ruth's age in 30 years' time is half as much again as Ann's in 12 years' time.
-17 Now +12 +30
Ruth 37 54 66 84
Ann 27 44 56 74
