SOLUTION: Hi paul put some beads in jars A B C.the ratio of the number of beads in A to B 2 to 3. The ratio of beads inB to C is2 to 1. If paul transfers an equal number of beads from B

It's solved below, under my pseudonym, AnlytcPhil :)



Edwin 

Suppose in the beginning,
 
jar A contains "a" beads,
jar B contains "b" beads
jar C contains "c" beads

Let's do this in steps so that we don't get mixed up:

1. He takes n beads out of B, so B now contains only b-n beads.
2. He then puts those n beads in A, so A now contains a+n beads.
3. He then takes n more beads out of B, so B now contains only 
   b-2n beads. (He took another n beads out of B).
4. He puts those n beads in C, so C now contains c+n beads.

Now A and B have the same number of beads, so we set them
equal:

So we have 4 equations in 4 unknown

b appears in 3 of the equations, so we substitute
2c for b in the 1st, 3rd and 4th:





c occurs in all three equations, so we solve the second 
for c = 297-n and substitute 297-n for c in the first two
equations:







Solve the second equation for a = 594-5n and
substitute 594-5n for a in the first equation









Substitute 54 for n in






Also substitute 54 for n in 





Substitute 243 for c in






So A contained a=324 beads\
B contained b=486 beads
C contained c=243 beads

Checking:
 
1. When he took n=54 beads from B, B then contained only 
   486-54=432 beads.
2. When he put those 54 beads in A, A then contained 
   324+54=378 beads. 378.
3. When he took n=54 more beads out of B, B then contained 
   only 432-54=378 beads.

Aha!  That checks for A and B then both contained the same
number of beads,  

To finish,

4. When he put those n=54 beads in C, C then contained 
   243+54=297 beads.

That checks.  So we're right.  

Edwin