System of Weights

Question

The following figure shows a system of weights that is perfectly balanced. The total weight is 192 kg. What's the weight of the purple square?

Accepted Answer

Let's define the following:

 	G = Green triangle
 	P = Purple square
 	L = Lightblue star
 	B = Blue cloud
 	O = Orange flag
 	C = Grey circle
 	R = Red turned square

Let's now define the following system of equations: \begin{equation} 2G + 2P + 2L + B = R + G + 2C + P + 2O = 96 \end{equation} From the provided figure, we can also define the following: \begin{equation} P = 2O \end{equation} \begin{equation} G + C = P + 2O = 4O \end{equation} \begin{equation} G + C = P + 2O = 4O = 24 \end{equation} \begin{equation} R + C = 48 \end{equation} If you subtract the last two equations from each other, we get \begin{equation} R - G = 24 \end{equation} \begin{equation} R = 24 + G \end{equation} We also now know that \begin{equation} O = \frac{24}{4} = 6 \end{equation} We can substitute these in \begin{equation} R + G + 2C + P + 2O = 96 \end{equation} which gives us \begin{equation} (24+G) + G + 2C + P + (2*6) = 96 \end{equation} \begin{equation} 2G + 2C + P = 60 \end{equation} We already knew that \begin{equation} G + C  = 24 \end{equation} So, subtracting this twice gives us \begin{equation} 2G + 2C + P - 2*(G+C) = 60 - 2*24 \end{equation} \begin{equation} P = 12 \end{equation} We can sanity check this, because P should be equal to 2 times O, which is correct.

EXTRA PROOF
We already have our answer (P = 12), but we can continue from here, by solving all elements.


From the left side of the system, we know that \begin{equation} \frac{96-G}{2} = G + 2P = G + 24 \end{equation} \begin{equation} 24 = \frac{3G}{2} \end{equation} \begin{equation} G = 16 \end{equation} We stated earlier that \begin{equation} G + C = P + 2O = 4O \end{equation} So, after filling in G and O we now know that \begin{equation} 16 + C = 24 \end{equation} \begin{equation} C = 8 \end{equation} We also knew that \begin{equation} R = 24 + G \end{equation} So, we now know that \begin{equation} R = 40 \end{equation}  We can see that \begin{equation} B = 2L \end{equation} \begin{equation} B + 2L = \frac{96-G}{2} = \frac{96-16}{2} = 40 = 4L \end{equation} This results in \begin{equation} L = 10 \end{equation} and \begin{equation} B = 2L = 20 \end{equation} So finally, we solved all elements:

 	G = 16
 	P = 12
 	L = 10
 	B = 20
 	R = 40
 	C = 8
 	P = 12
 	O = 6

We can now check if this is correct by filling in \begin{equation} 2G + 2P + 2L + B = R + G + 2C + P + 2O = 96 \end{equation} \begin{equation} 32 + 24 + 20 + 20 = 40 + 16 + 16 + 12 + 12 = 96 \end{equation} \begin{equation} 96 = 96 = 96 \end{equation} So, all correct!

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