Two types of ore

A total of 42 tons of two types of ore is to be added into a smelter. The first type contains 6% copper and the other contains 2.5% copper. Find the necessary amounts of each ore to produce 2 tons of copper.

Correct answer:

a =  27.1429 t
b =  14.8571 t

Step-by-step explanation:

$$\begin{gathered} r_1=6\%=\frac{6}{100}=\frac{3}{50}=0.06 \\ {r}_2=2.5\%=\frac{2.5}{100}=0.025 \\ {m}_1=42\text{t} \\ {m}_2=2\text{t} \\ a+b=m_1 \\ {r}_1\cdot a+r_2\cdot b=m_2 \\ a+b=42 \\ 0.06\cdot a+0.025\cdot b=2 \\ a+b=42 \\ 0.06a+0.025b=2 \\ Row2-0.06\cdot Row1\to Row2 \\ a+b=42 \\ -0.04b=-0.52 \\ b=\frac{-0.52}{-0.035}=14.85714286 \\ a=42-b=42-14.85714286=27.14285714=27.1429\text{t} \\ a=\frac{190}{7}\doteq 27.142857 \\ b=\frac{104}{7}\doteq 14.857143 \end{gathered}$$

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