Cashews + beetroots

The lady sells nuts. It has two mixtures of nuts (cashews/beetroots) in a ratio of 2:1 and 1:3 (cashews and peanuts). Determine the ratio to mix both mixtures so that the ratio of cashews/beetroot in the resulting mixture is 1:2.
(Please provide a certificate on which principle this is calculated)

Final Answer:

r =  1:4

Step-by-step explanation:

$$\begin{gathered} s_1:k:b=2:1=2x:1x \\ {s}_2:k:b=1:3=1y:3y \\ {s}_1:k=2b \\ {s}_2:k=b/3 \\ k:b=1:2 \\ x=1 \\ k=2\cdot x+1\cdot y \\ b=1\cdot x+3\cdot y \\ k=b/2 \\ k=2\cdot 1+1\cdot y \\ b=1\cdot 1+3\cdot y \\ k=b/2 \\ k-y=2 \\ b-3y=1 \\ b-2k=0 \\ Pivot:Row1\leftrightarrow Row2 \\ Row3-Row1\to Row3 \\ -2k+3y=-1 \\ Pivot:Row2\leftrightarrow Row3 \\ Row3-\frac{1}{-2}\cdot Row2\to Row3 \\ 0.5y=1.5 \\ y=\frac{1.5}{0.5}=3 \\ k=\frac{-1-3y}{-2}=\frac{-1-3\cdot 3}{-2}=5 \\ b=1+3y=1+3\cdot 3=10 \\ b=10 \\ k=5 \\ y=3 \\ r=\frac{2\cdot x+1\cdot x}{1\cdot y+3\cdot y}=\frac{2\cdot 1+1\cdot 1}{1\cdot 3+3\cdot 3}=\frac{1}{4}=0.25=1:4 \end{gathered}$$

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