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To prepare 14 kg of cheese with 45% dry matter, 3 types of cheese mixtures were mixed. The first was 5 kg, the second was 3 kg and had 14% more dry matter (14 percentage points more) than the first. The third mixture, which had 2 times as much dry matter as the first mixture, was 6 kg. What % dry matter was in each cheese mixture?

Correct answer:

p₁ =  29.4 %
p₂ =  43.4 %
p₃ =  58.8 %

Step-by-step explanation:

$$\begin{gathered} r_1=14\%=\frac{14}{100}=\frac{7}{50}=0.14 \\ {r}_2=45\%=\frac{45}{100}=\frac{9}{20}=0.45 \\ {m}_1=5\text{kg} \\ {m}_2=3\text{kg} \\ {m}_3=6\text{kg} \\ m=m_1+m_2+m_3=5+3+6=14\text{kg} \\ S=m\cdot r_2=14\cdot 0.45=\frac{63}{10}=6.3\text{kg} \\ {m}_1\cdot r+m_2\cdot (r+r_1)+m_3\cdot 2\cdot r=S \\ 5\cdot r+3\cdot (r+0.14)+6\cdot 2\cdot r=6.3 \\ 20r=5.88 \\ r=\frac{5.88}{20}=0.294 \\ r=\frac{147}{500}=0.294 \\ {p}_1=100\cdot r=100\cdot 0.294=29.4\% \end{gathered}$$

$p_2=100\cdot (r+r_1)=100\cdot (0.294+0.14)=43.4\%$

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