GMAT Challenge Series (700+): Q7

Alloy.jpg

Question:

An alloy of copper and aluminum has 40% copper. An alloy of Copper and Zinc has Copper and Zinc in the ratio 2:7. These two alloys are mixed in such a way that in the overall alloy, there is more aluminum than Zinc, and copper constitutes x% of this alloy. What is the range of values x can take?

A. 30% ≤ x ≤ 40%

B. 32.5% ≤ x ≤ 40%

C. 32.5% ≤ x ≤ 42%

D. 33.33% ≤ x ≤ 40%

E. 33.33 % ≤ x ≤ 42%

MBA Wisdom's Answer:

$$\text{Alloy 1}$$ $$\textit{Aluminium} = \text{60%}$$ $$\textit{Copper} = \text{40%}$$ $$\textit{Zinc} = \text{0%}$$ $$ $$ $$\text{Alloy 1}$$ $$\textit{Aluminium} = \text{0%}$$ $$\textit{Copper} = \frac{2}{9} \approx \text{22%}$$ $$\textit{Zinc} = \frac{2}{9} \approx \text{78%}$$ $$ $$ $$\text{Mixture of Alloy 1 and Alloy 2}$$ $$\textit{Let y equal the % of Alloy 1 in the mixture}$$ $$\textit{Aluminum in the mixture} = \left( \frac{3}{5} \right) \textit{y}$$ $$\textit{Zinc in the mixture} = \left( \frac{7}{9} \right) \left( {1} - \textit{y} \right)$$ $$\textit{Aluminum in the mixture} > \textit{Zinc in the mixture}$$ $$\left( \frac{3}{5} \right) \textit{x} > \left( \frac{7}{9} \right) \left( {1} - \textit{y} \right)$$ $$\left( \frac{54}{90} \right) \textit{x} > \left( \frac{7}{9} \right) \textit{(1-y)}$$ $${124}\textit{y} > {70}$$ $$\textit{y} > \frac{35}{62}$$ $$$$ $$\textit{Lowest value of x is when y} = \frac{35}{62}$$ $$\textit{Lowest value of x} = \left( \frac{35}{62} \right)\left( \frac{40}{100} \right) + \left( \frac{27}{62} \right)\left( \frac{2}{9} \right)$$ $$\textit{Lowest value of x} = \frac{14}{62} \frac{6}{62}$$ $$\textit{Lowest value of x} = \frac{20}{62} \approx \textit{32.26%}$$ $$$$ $$\textit{Highest value of x is when y equals 100% in which case x equals 40%.}$$ $$$$ $${\textit{Best range: 32.5%} \leq \textit{x}} \leq \textit{40%}$$

B. 32.5% ≤ x ≤ 40%