An alloy (X) of gold and silver is taken in the ratio 1 : 2 and another alloy (Y) of gold and silver is taken in the ratio 2 : 3. How many parts of X and Y must be taken to obtain a new alloy Z consisting of gold and silver in the ratio 4 : 7?
6 : 5
In alloy X, gold fraction \(=\frac{1}{3}\) and silver \(=\frac{2}{3}\); in alloy Y, gold fraction \(=\frac{2}{5}\) and silver \(=\frac{3}{5}\). Mixing \(x\) parts of X and \(y\) parts of Y to get gold : silver \(=4:7\): \(7\left(\frac{x}{3}+\frac{2y}{5}\right)=4\left(\frac{2x}{3}+\frac{3y}{5}\right)\). Simplifying gives \(\frac{2y}{5}=\frac{x}{3} \Rightarrow x:y=6:5\).
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