BW-BS balls

Adam has a full box of large or small balls, black or white. The ratio between large and small balls is 5:3. Within the large balls, the ratio of black to white is 1:2, and between small balls, the ratio of black to white is 1:8.

What is the ratio of all-black to all-white balls?

Final Answer:

x =  5:3

Step-by-step explanation:

$$\begin{gathered} a=1 \\ a+b=5/3\cdot (c+d) \\ a=1/2\cdot b \\ c=1/8\cdot d \\ 1+b=5/3\cdot (c+d) \\ 1=1/2\cdot b \\ 3b-5c-5d=-3 \\ b=2 \\ 8c-d=0 \\ Row2-\frac{1}{3}\cdot Row1\to Row2 \\ 1.667c+1.667d=3 \\ Pivot:Row2\leftrightarrow Row3 \\ Row3-\frac{1.66666667}{8}\cdot Row2\to Row3 \\ 1.875d=3 \\ d=\frac{3}{1.875}=1.6 \\ c=\frac{0+d}{8}=\frac{0+1.6}{8}=0.2 \\ b=\frac{-3+5c+5d}{3}=\frac{-3+5\cdot 0.2+5\cdot 1.6}{3}=2 \\ c=\frac{1}{5}=0.2 \\ d=\frac{8}{5}=1.6 \\ x=\frac{a+b}{c+d}=\frac{1+2}{0.2+1.6}\approx 1.6667=5:3 \end{gathered}$$

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