Bottom line: J = 8, M = 15, T = 3
BTW, This problem beautifully demonstrates why students tend to hate word problems.
Also, note, my 1st post had some arithmetic errors. This one is solid.
Let T, J, and M be their current ages.
(All the derivations below are done with slow simple steps to avoid errors)
Statement 1, Introduces: P1
-----------------------------------
10 years from now Toby will be twice as old
as Julie was when... (let P1 be the number of years ago)
T + 10 = 2(J-P1)
...Melanie was 9 times as old as Toby.
M - P1 = 9(T - P1)
M - P1 = 9T - 9P1
8P1 = 9T - M
P1 = (9T - M)/8
Going back:
T + 10 = 2(J - [9T - M]/8)
8T + 80 = 16J - 18T + 2M
16J + 2M - 26T = 80
-----------------------
8J + M - 13T = 40
-----------------------
Statement 2, Introduces: F1, F2
--------------------------------
Let the first "will be" be F1 years in the future.
Let the 2nd and 3rd "will be"s be F2 years in the future.
(The 4th "will be" is 2 years explicitly 2 years from now.)
8 years ago, Melanie was half as old as Julie will be...
2(M - 8) = J + F1,
...when Julie is 1 year older than Toby will be...
J + F1 = 1 + (T + F2)
(First, get eqn. in terms of F1)
F1 = -J + T + F2 + 1
.... at the time when Melanie will be 5 times
as old as Toby will be 2 years from now.
M + F2 = 5(T + 2)
F2 = -M + 5T + 10
Substituting into expression for F1:
F1 = -J + T - M + 5T + 10 + 1
F1 = -J - M + 6T + 11
And, going back to the 1st eqn:
2(M - 8) = J - J - M + 6T + 11
2M - 16 = -M + 6T + 11
----------------
3M - 6T = 27
----------------
Statement 3, Introduces: F3, P2, F4, P3, F5
--------------------------------------------
When Toby was 1 year old, ... (T - 1 years ago)
Melanie was 3 years older than...
M - (T - 1) = 3 + ...
M - T + 1 = 3 + ....
...Toby will be when... ("when" is F3 years in the future)
M - T + 1 = 3 + (T + F3)
F3 = M - 2T - 2
...Julie is 3 times as old as Melanie was 6 years before
the time when... (where "the time when" is P2 years before now)
J + F3 = 3(M - P2 - 6)
F3 = -J + 3M - 18 - 3 P2
...when Julie was half as old as Toby will be when ("when"
here is F4 years into the future)
2(J - P2) = T + F4
2 P2 = 2J - T - F4
...Melanie will be 10 years older than Melanie was... (P3 years ago)
M + F4 = 10 + (M - P3)
F4 = 10 - P3
...when Julie was one-third as old as Toby will be
(J - P3) = (1/3) (T + F5)
3(J - P3) = T + F5
3 P3 = 3J - T - F5
...when Melanie will be 3 times as old as she was when Julie was born.
M + F5 = 3(M - J)
F5 = 2M - 3J
Working backwards starting from three lines before:
3 P3 = 3J - T - F5
3 P3 = 3J - T - (2M - 3J)
3 P3 = 3J + 3J - 2M - T
3 P3 = 6J - 2M - T
P3 = 2J - (2/3)M - (1/3)T
F4 = 10 - P3
F4 = 10 -2J + (2/3)M + (1/3)T
F4 = -2J + (2/3)M + (1/3)T + 10
2 P2 = 2J - T - F4
2 P2 = 2J - T + 2J - (2/3)M - (1/3)T - 10
2 P2 = 4J - (2/3)M - (4/3)T - 10
P2 = 2J - (1/3)M - (2/3)T - 5
F3 = -J + 3M - 18 - 3P2
F3 = -J + 3M - 18 - 6J + M + 2T + 15
F3 = -7J + 4 M + 2T - 3
F3 = M - 2T - 2
F3 = F3
M - 2T - 2 = -7J + 4 M + 2T - 3
-------------------------
7J - 3M - 4T = -1
-------------------------
(Charlie got this last one, and did so more directly.)
So:
8J + M - 13T = 40
3M - 6T = 27
7J - 3M - 4T = -1
(for this - rather than solving the system with determinants or substitution, I just plugged it into Wolfram Alpha.)
=============================
Gives: J = 8 M = 15 T = 3
============================
Here is the check:
J M T P1 P2 P3 F1 F2 F3 F4 F5
-----------------------------------------------------------------
8.0 15.0 3.0 1.5 4.0 5.0 6.0 10.0 7.0 5.0 6.0
Statement 1 Check
------------------------------
10 years from now, old-aged T will be 13.0
1.5 years ago, M was 13.5, which is 9 times what T was then: 1.5
1.5 years ago J was 6.5, half the age of old-aged T.
Statement 2 Check
------------------------------
In 2 yrs T is 5.0. In 10.0 yrs, M will be 25.0 yrs, five times older.
At that time, 10.0 from now, T will be 13.0
and 6.0 years from now, J will be will one year
older than that age, at 14.0 years
8 years ago M was 7.0 which is half of
J's previously quoted age.
Statement 3 Check
------------------------------
In 6.0 yrs, M will be 21.0, which is 3 x 7.0 (her age at J's birth)
At that time, 6.0 years from now, T will be 9.0
and 5.0 years ago J was 3.0, which is (1/3) T's future age.
5.0 years ago, M was the age 10.0 years old.
In 5.0 years, she will be 20.0, 10 years older.
5.0 years from now, T will be 8.0.
4.0 years ago, J was 4.0 which is half T's future age.
4.0 and 6 more years ago, M was 5.0. That is
(1/3) as old as J will be (15.0) in 7.0 years.
2.0 years ago, when T was 1 year old,
M was 13.0 years old. Her age then was 3 years less
than T will be 7.0 years from now at age 10.0
(I think this might be enough to discourage some folks from having kids...)
Edited on January 11, 2022, 7:07 pm
soln
