Bottom line: J = 8, M = 15, T = 3

BTW, This problem beautifully demonstrates why students tend to hate word* *problem*s*.

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**