In the picture below, same objects have the same weight and different objects have different weights.?

Question

In the picture below, same objects have the same weight and different objects have different weights. Scales A, B, and C give the total weight in pounds for the objects above them. How many pounds does one cube weigh?
click below for the picture:
http://img443.imageshack.us/my.php?image=mathiq6.png

Answer 1

Let's assign values for each of the objects:
x = weight of the cylinder
y = weight of the sphere
z = weight of the cube

From the 3 scales we know:
A) 2x + 2y + z = 67
B) x + 3y + z = 68
C) 2x + y = 43

So you have 3 equations and 3 unknowns:
Let's double equation B so it has 2x just like the other equations:
B) 2x + 6y + 2z = 136

Now you can subtract A from B to eliminate the x variable.
2x + 6y + 2z = 136
2x + 2y + z = 67
4y + z = 69

And subtract C from A to also eliminate x:
2x + 2y + z = 67
2x + y = 43
y + z = 24

Now you have two equations and two unknowns.
4y + z = 69
y + z = 24

Multiply the last equation by 4 so that you have the same 4y at the beginning.
4y + 4z = 96

Subtract the prior equation to eliminate the y variable:
4y + 4z = 96
4y + z = 69

4y + 4z - (4y + z) = (96 - 69)
4z - z = 27
3z = 27
z = 9

Thus the cube weighs 9 lbs.

You can also figure the weights of the other shapes by substituting them back into the equations:
y + z = 24
y + (9) = 24
y = 15

2x + y = 43
2x + (15) = 43
2x = 28
x = 14

Putting it all together:
Cylinder weighs 14 lbs.
Sphere weighs 15 lbs.
Cube weighs 9 lbs.

Or if you want to do it intutively...

Notice that the only difference between scale A and B is the sphere. This means a sphere must weigh 1 lb. more than a cylinder.

Now if you imagine replacing the sphere on scale C with a cylinder it should go down to 42 lbs. 3 cylinders at 42 lbs. means each one is 14 lbs.

And the sphere is 1 lb. heavier or 15 lbs.

Finally you can easily figure out that the cube is 9 lbs.

Answer 2

♣ comparing A & B you conclude: CAN + 1 = BALL;
♦ looking at C you write the equation: 2*CAN + BALL = 43 lb, thence
2*CAN + CAN + 1 =43, hence CAN =14, BALL = 15;
♥ therefore returning to A you conclude: 2*CAN + 2*BALL + CUBE = 67, thence 2*14 +2*15 + CUBE = 67, hence CUBE = 9;
looking at B we check: 14 +3*15 +9 =68;