Say you have the following target proportion:
$$3:4:5$$
and you are also given a few other proportions:
$$1:2:3$$
$$3:7:1$$
$$2:1:2$$
with the task of scaling those 3 proportions so their ratio is equivalent to the first proportion given.
for this example you multiply the first by 8, the second by 1, and the third by 5 to give you:
$$8:16:24$$
$$3:7:1$$
$$10:5:10$$
which when you add all the columns together gives you:
$$21:28:35$$
which matches the target proportion of $3:4:5$. My only issue is how you go about mathematically finding which numbers to multiply each proportion by, as opposed to just plugging random numbers in and hoping a combination of them eventually works. I can tell that they all have a common GCF of 7 and looking at this example and another example, it seems that the proportion that gets multiplied by 1 is the proportion that gives you the largest number when you multiply all the parts of it together.
Best Answer
If you multiply the first ratio $(1:2:3)$ by $a$, the second by $b$, and the third by $c$, then sum these ratios together, you get $a+3b+2c : 2a + 7b + c : 3a + b + 2 c$, and you are finding $a,b,c$ such that this ratio is $3 : 4 : 5$.
Therefore you have three equations in three unknowns below:
$$a + 3b + 2c = 3$$ $$2a + 7b + c = 4$$ $$3a + b + 2c = 5$$
where $a,b,c$ are not necessarily integers.
You can use matrices to solve for this system, or you can just isolate one of the variables, bringing it down to two equations in two unknowns if you want to do this by hand.
Alternatively, a GDC (graphics display calculator) or WolframAlpha will solve the system of equations for you, giving $(a, b, c) = (\frac{8}{7}, \frac{1}{7}, \frac{5}{7})$. Since the common denominator is $7$, multiplying $3:4:5$ by $7$ gives $21:28:35$.