Given Problem:
A manufacturing process mixes three raw materials $M_1$, $M_2$, and $M_3$ to produce three products $P_1$, $P_2$, and $P_3$. The ratios of the amounts of the raw materials (in the order $M_1$, $M_2$, $M_3$) which are used to make up each of the three products are as follows: for $P_1$ the ratio is $1:2:3$; for $P_2$ the ratio is $1:3:5$; and for $P_3$ the ratio is $3:5:8$. In a certain production run, $137$ units of $M_1$, $279$ units of $M_2$, and $448$ units of $M_3$ were used. The problem is to determine how many units of each of the products $P_1$, $P_2$, and $P_3$ were produced in that run. a) Set this problem up in matrix form. Use the letter $A$ for the matrix, and write down the (one-line) formula for the solution in matrix form.
In the given solution to this problem, the first statement they make is:
In $1$ unit of $P_1$ there are $\frac{1}{6}$ of a unit of $M_1$, $\frac{2}{6}$ of a unit of $M_2$, and $\frac{3}{6}$ of a unit of $M_3$.
I am confused as to how they reached this conclusion from the given statements in the problem.
If you use materials in the ratio $1:2:3$, note first that $1+2+3=6$, so $1/6$ of the product is material $1$, $2/6 = 1/3$ of the product is material $2$, and $3/6=1/2$ is material $3$. They're measuring everything in units (say grams). $1$ gram of the first product consists of $1/6$ grams of material $1$, $1/3$ grams of material $2$, and $1/2$ gram of material $3$.