Determine the present ages of each of the three siblings Toby, Julie and Melanie from the following clues:

1. 10 years from now Toby will be twice as old as Julie was when Melanie was 9 times as old as Toby.
2. 8 years ago, Melanie was half as old as Julie will be when Julie is 1 year older than Toby will be at the time when Melanie will be 5 times as old as Toby will be 2 years from now.
3. When Toby was 1 year old, Melanie was 3 years older than Toby will be when Julie is 3 times as old as Melanie was 6 years before the time when Julie was half as old as Toby will be when Melanie will be 10 years older than Melanie was when Julie was one-third as old as Toby will be when Melanie will be 3 times as old as she was when Julie was born.

1.
Nine times Toby's age                  9T
When was Melany that age               9T - M
How old was Julie then                 9T - M + J
Twice that                             18T - 2M + 2J
When will Toby be that age             17T - 2M + 2J
This is said to be at t=10
           10 = 17T - 2M + 2J
2.
Toby's age 2 years from now            T+2
Five times that                        5T + 10
When will Melanie be that age          5T + 10 - M
Toby's age at that time                6T + 10 - M
One year older than that               6T + 11 - M
When will Julie be that age            6T + 11 - M - J
What's half that                       3T + 11/2 - M/2 - J/2
When is Melanie that                   3T + 11/2 - 3M/2 - J/2
That's said to be at t = - 8
          -8 = 3T + 11/2 - 3M/2 - J/2

3. 
Julie's age now (t=0)                  J
Melanies age at Julie's birth          M-J
Three times that                       3M-3J
When will she be 3 times that          3M-3J-M = 2M-3J
                                      (since now is t=0)
How old will Toby be then              T+2M-3J
One third of that                      (T+2M)/3 - J
When was Julie that age                (T+2M)/3 - 2J
                                      (should be a negative
                                        value relative to t=0)
How old was Melanie then               M+(T+2M)/3 - 2J  
                                      (adding a negative value)
                                       = T/3 + 5M/3 - 2J
Ten years older than that              T/3 + 5M/3 - 2J + 10                                       
When will Melanie be that              T/3 + 2M/3 - 2J + 10                         
How old will Toby be then              4T/3 + 2M/3 - 2J + 10
What's half that                       2T/3 + M/3 - J + 5
When was Julie that age                2T/3 + M/3 - 2J + 5
Six years before that was t=           2T/3 + M/3 - 2J - 1
How old was Melanie then               2T/3 + 4M/3 - 2J - 1
What's 3 times that                    2T + 4M - 6J - 3
When will Julie be that age            2T + 4M - 7J - 3
How old will Toby be then              3T + 4M - 7J - 3
Three years older than that            3T + 4M - 7J
When was Melanie that age              3T + 3M - 7J (this is negative, 
                                                         i.e. the past)
Toby was 1 then so now he's            1 - 3T - 3M + 7J
              T = 1 - 3T - 3M + 7J
                      or
               4T = 1 - 3M + 7J  

Posted by Charlie on 2022-01-08 18:11:58