How to make logical sense of following problem with ratios/percentages

Question

I have the following variables

L	M	T
L1	M1	T1
L2	M2	T2
L3	M3	T3
L4	M4	T4

where $L+M=T$

I want to calculate an approximate value for the 4s using the ratio between the 2s and 1s, and then multiply it with the 3s.

To make it clear: $\frac{T2}{T1}T3=T4$

But, this is using only the totals. You could also do it one by one: $\frac{L2}{L1}L3=L4$ etc.

Though, doing it this way results in $L4+M4\neq T4$

The math checks out for this, but I struggle to make sense of it logically. Why does this make sense from a logical perspective, and which way of doing it would be correct for my purposes?

Answer 1

I believe you wish that this was true:

$$\frac{L_2}{L_1}L_3+\frac{M_2}{M_1}M_3=\frac{T_2}{T_1}T_3$$

which is:

$$\frac{L_2}{L_1}L_3+\frac{M_2}{M_1}M_3=\frac{L_2+M_2}{L_1+M_1}(L_3+M_3)$$

which can be transformed to:

$$(L_1M_2-L_2M_1)(L_1M_3-L_3M_1)=0$$

(see Wolfram Alpha), which is:

$$\det\begin{bmatrix}L_1&M_1\\L_2&M_2\end{bmatrix}\cdot \det\begin{bmatrix}L_1&M_1\\L_3&M_3\end{bmatrix}=0$$

Now, the determinant of a matrix is sort-of a measure how linearly independent its rows or columns are. It is closely related to the volume of the parallelepiped spanned by those rows, or columns. For $2\times 2$ matrices, the parallelepiped is just the parallelogram, the "volume" is just the area, and the determinant is the measure of how far those rows or columns are from being proportional to each other. (If proportional, the parallelogram becomes degenerate and its area is $0$.)

So this explains why the belief that the above is true sounds so "logical". It will be true if either $(L_1, M_1)$ and $(L_2, M_2)$ are proportional to each other, or if $(L_1, M_1)$ and $(L_3, M_3)$ are proportional to each other. Which often happens if you are observing some trend in real life.

And, of course, if neither of those pairs are proportional to each other, then the first formula above is, sadly, false.

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I will refrain from answering your last question:

... which way of doing it would be correct for my purposes?

as I don't know what your purpose is, but you have to give up either $T_4=\frac{T_2}{T_1}T_3$ or $T_4=L_4+M_4$. You choose which one. You may preserve the other.