Preparing the mixture

The seller prepared 25 kg of the mixture for 264 CZK per 1 kg. The first type was 180 CZK per 1 kg and the second was 390 CZK per kg. How much of each did he need?

Correct answer:

a =  15 kg
b =  10 kg

Step-by-step explanation:

$$\begin{gathered} m=25\text{kg} \\ m=a+b \\ c=264\text{CZK/kg} \\ {c}_1=180\text{CZK/kg} \\ {c}_2=390\text{CZK/kg} \\ a+b=m \\ {c}_1\cdot a+c_2\cdot b=c\cdot m \\ a+b=25 \\ 180\cdot a+390\cdot b=264\cdot 25 \\ a+b=25 \\ 180a+390b=6600 \\ Pivot:Row1\leftrightarrow Row2 \\ 180a+390b=6600 \\ a+b=25 \\ Row2-\frac{1}{180}\cdot Row1\to Row2 \\ 180a+390b=6600 \\ -1.17b=-11.67 \\ b=\frac{-11.66666667}{-1.16666667}=10 \\ a=\frac{6600-390b}{180}=\frac{6600-390\cdot 10}{180}=15=15\text{kg} \\ a=15 \\ b=10 \end{gathered}$$

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