There were three brothers Albert, Bertrand and Charles.

Albert lent to Bertrand and Charles as many books as they had initially.

After some time Bertrand gave as many books to Albert and Charles as many as they now have.

After sometime Charles did the same thing.

At the end each one of them had 24 books.

Find the books each orginally had.

Call a the number of books Albert started with, b the number of books Bertrand started with, and c the number of books Charles started with. Since they do not buy or throw out any books, a+b+c=3(24)=72 (Equation 0)

Initial State (State 0):
Albert a
Bertrand b
Charles c


Albert lends books out (State 1):
Albert a-b-c
Bertrand 2b
Charles 2c

Bertrand lends books out (State 2):
Albert 2(a-b-c) = 2a-2b-2c
Bertrand 2b-(a-b-c)-2c = 3b-a-c
Charles 4c

Charles lends his books out (State 3):
Albert 2(2a-2b-2c)=4a-4b-4c=24 -> a-b-c=6 (Equation 1)
Bertrand 2(3b-a-c)=6b-2a-2c=24 -> 3b-a-c=12 (Equation 2)
Charles 4c-(2a-2b-2c)-(3b-a-c)=7c-b-a=24 (Equation 3)

Adding Equations 1 and 3:
 a-b-c=06
-a-b+7c=24
___________
 -2b+6c=30 -> 3c-b=15 (Equation 4)

Comparing Equation 1 with State 1, we see that Albert has 6 books, so Bertrand and Charles must have 66 between them: 2b+2c=66 -> b+c=33 (Equation 5)

Adding Equations 4 and 5
3c-b=15
 c+b=33
_________
4c  =48 -> c=12

Substituting in Equation 5: b+12=33 -> b=21

So Albert started out with 39 books, Bertrand with 21 books and Charles with 12 books.

State 0: Albert 39, Bertrand 21, Charles 12
State 1: Albert 06, Bertrand 42, Charles 24
State 2: Albert 12, Bertrand 12, Charles 48
State 3: Albert 25, Bertrand 24, Charles 24

Posted by TomM on 2003-02-02 07:08:42