Alloy

The first alloy is a mixture of two metals in the ratio 1:2; the second is a mixture of the same metals in the ratio 2:3. In what ratio must these two alloys be combined in the furnace to obtain a new alloy with a ratio of 17:27? (All three ratios refer to the same order of the two metals.)

Final Answer:

r = 9:35

Step-by-step explanation:

k_{1} = \frac{1}{1 + 2} \cdot m_{1} + \frac{2}{2 + 3} \cdot m_{2} k_{2} = \frac{2}{1 + 2} \cdot m_{1} + \frac{3}{2 + 3} \cdot m_{2} k_{1} = \frac{1}{3} \cdot m_{1} + \frac{2}{5} \cdot m_{2} k_{2} = \frac{2}{3} \cdot m_{1} + \frac{3}{5} \cdot m_{2} k_{1} / k_{2} = 17 : 27 r = m_{1} / m_{2} 17 / 27 = \frac{\frac{1}{3} \cdot m _{1} + \frac{2}{5} \cdot m _{2}}{\frac{2}{3} \cdot m _{1} + \frac{3}{5} \cdot m _{2}} 17 \cdot (\frac{2}{3} \cdot m_{1} + \frac{3}{5} \cdot m_{2}) = 27 \cdot (\frac{1}{3} \cdot m_{1} + \frac{2}{5} \cdot m_{2}) 17 \cdot (\frac{2}{3} \cdot r + \frac{3}{5}) = 27 \cdot (\frac{1}{3} \cdot r + \frac{2}{5}) r = \frac{2 7 \cdot \frac{2}{5} - 1 7 \cdot \frac{3}{5}}{1 7 \cdot \frac{2}{3} - 2 7 \cdot \frac{1}{3}} = \frac{9}{3 5} ≐ 0.2571 Verifying Solution: m_{1} = 9 kg m_{2} = m_{1} / r = 9 / 0.2571 = 35 kg k_{1} = \frac{1}{3} \cdot m_{1} + \frac{2}{5} \cdot m_{2} = \frac{1}{3} \cdot 9 + \frac{2}{5} \cdot 35 = 17 kg k_{2} = \frac{2}{3} \cdot m_{1} + \frac{3}{5} \cdot m_{2} = \frac{2}{3} \cdot 9 + \frac{3}{5} \cdot 35 = 27 kg t_{2} = k_{1} / k_{2} = 17 / 27 = \frac{1 7}{2 7} ≐ 0.6296

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You need to know the following knowledge to solve this word math problem:

algebra

system of equations

basic operations and concepts

ratio

Units of physical quantities

mass

themes, topics

mixtures and solutions

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