Answer:
The degree that a given weight tends to pull down the left side is
proportional to the product of the weight and its distance from the
fulcrum. (You know this intuitively when playing on a see-saw with
someone whose weight is different from yours -- the heavier person has to
sit closer to the fulcrum so that the product of each person's weight and
their distance from the fulcrum equals that of the other person.)
Note that "A" is 7 units from the fulcrum.
"B" is 6 units, and "C" is 4 units. So the
"A" weight is pushing down with 8*7 = 56 units of rotational
force (properly called "torque"), the "B" weight is
pushing down with 5*6=30 units of torque, and the "C" weight is
pushing down with 15*4=60 units of torque.
On the right side, the 10-weight that's three units from the fulcrum is
pushing down with 30 units of torque, and the 4-weight that's 7 units from
the fulcrum is pushing down with 28 units of torque.
To see the effect of moving a weight, cover it with your hand, and look
at the remaining torque on the left side (the top numbers). Then add
four times the weight to the 58 known units of torque on the right side
(because the question mark is four units from the fulcrum).
So if the "A" weight is moved, 90 units of torque will be
left on the left, and it adds 8*4=32 units of torque to the 58 already on
the right side, for a total of 90 units on the right side. So the
scale will balance if A is moved in its entirety.
For completeness, let's see if the other weights will also balance the
scale.
If "B" is moved, 116 units will remain on the left side, and
5*4=20 units will be added to the right, for a total of 78 units on the
right side. Even if partial movement of 1, 2, 3, or 4 of the weights
from B are allowed, the two sides are never even.
If "C" is moved, 86 units will remain on the left side, and
15*4=60 units will be added to the right, for a total of 118 units on the
right side. But if only 11 of the 15 weights are moved from
"B" to "?", then there will be 102 units of torque on
each side.
So there are two answers...
The first answer:
For part a, the answer is that we moved the "A" stack, 8
pounds.
For part b, the answer is that 20 pounds remain on the left side, and
22 pounds are on the right side.
The second answer:
For part a, the answer is that we moved 11 pounds from the "C"
stack.
For part b, the answer is that 17 pounds remain on the left side, and
25 pounds are on the right side.
Either answer should be counted as correct.
