Statement 1
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10 years from now Toby will be twice as old
T + 10 = 2(...
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
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8J + M - 13T = 40
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Statement 2
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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.
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
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3M - 6T = 27
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Statement 3
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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... (F3 years in the future)
M - T + 1 = 3 + (T + F3)
or,
F3 = M - 2T - 2
...Julie is 3 times as old as Melanie was 6 years before
J + F3 = 3(M - P2 - 6)
F3 = -J + 3M - 3P2 - 6
... the time (where "the time" is P2 years before now)
when Julie was half as old as Toby will be when ("when"
here is F4 years into the future)
J - P2 = 2(T + F4)
(get P2 on the left)
P2 = J - 2T - 2F4
...Melanie will be 10 years older than Melanie was
M + F4 = 10 + (M - P3)
F4 = 10 - P3
when Julie was one-third as old as Toby will be
J - P3 = 3(T + F5)
P3 = J - 3T - 3F5
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 four lines before:
P3 = J - 3T - 3F5 = J - 3T - 3(2M - 3J)
P3 = 10J - 6M - 3T
F4 = 10 - P3
F4 = -10J + 6M + 3T + 10
P2 = J - 2T -2F4
P2 = J - 2T + 20J - 12M - 6T - 20
P2 = 21J - 12M - 8T - 20
F3 = -J + 3M - 3P2 - 6
F3 = -J + 3M -63J + 36M + 24T + 60 - 6
F3 = -64J + 39M + 24T + 54
F3 = M - 2T - 2
-64J + 39M + 24T + 54 = M - 2T - 2
-64J + 38M + 26T = -56
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32J - 19M - 13T = -28
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So
8J + M - 13T = 40
3M - 6T = 27
32J - 19M - 13T = -28
Gives: J = 1/7 M = 25/7 T = -19/7
Clearly there are errors or misunderstandings or
most likely both.
my work so far
