Determine the present ages of each of the three siblings Toby, Julie and Melanie from the following clues:
 10 years from now Toby will be twice as old as Julie was when Melanie was 9 times as old as Toby.
 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.
 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 onethird as old as Toby will be when Melanie will be 3 times as old as she was when Julie was born.
Time now is t = 0
Julie is age J
Toby is age T
Melanie is age M
To get to a date when X (at age X now) reaches age Ad (destination age), take Ad  X.
To get to the age of X at time Td (destination time), take X + Td.
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 MJ
Three times that 3M3J
When will she be 3 times that 3M3JM = 2M3J
(since now is t=0)
How old will Toby be then T+2M3J
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
eqns =
[10 == 2*J  2*M + 17*T, 8 == 3*T  (3*M)/2  J/2 + 11/2, 4*T == 7*J  3*M + 1]
soln=solve(eqns,[T,M,J])
soln.T
ans =
58/49
>> soln.M
ans =
1919/196
>> soln.J
ans =
927/196
Wolfram Alpha agrees, as
solve 10 = 17T  2M + 2J, 8 = 3T + 11/2  3M/2  J/2, 4T = 1  3M + 7J for T,M,J
yields the same answers: 58/49 for T, 1919/196 for M and 927/196 for J.

Posted by Charlie
on 20220108 18:11:58 