paul put some beads in jars A B C.
Suppose in the beginning,
jar A contains "a" beads,
jar B contains "b" beads
jar C contains "c" beads
the ratio of the number of beads in A to B 2 to 3.
The ratio of beads in B to C is 2 to 1.
If paul transfers an equal number of beads from B to A
and C he will have an equal number of beads in A and B
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:
and the total of beads in C will increase to 297.
So we have 4 equations in 4 unknown
find the number of beads in all 3 jars.
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