# finding ratio of elements when 2 solids are melted to form another solid

Here is my question Suppose if a person has two solids.Solid A which is made up of two elements, element X and element Y and similarly there is also Solid B which is also made up of same two elements.A consists of elements X and Y in ratio 4:9 and B consists of element X and Y in ratio 5:6.Now if equal amount of A and B are melted to form another element C then how to find out the ratio of X and Y in C. Answer is 109:177

We give a marginally more concrete version of Simon Markett's answer. Imagine that we melt together $k$ kilograms of A and $k$ kilograms of B.

The resulting material has $\dfrac{4k}{13}+\dfrac{5k}{11}$ kilograms of X and $\dfrac{9k}{13}+\dfrac{6k}{11}$ kilograms of Y.

Note that
$$\frac{4k}{13}+\frac{5k}{11}=\frac{109k}{(13)(11)}\quad\text{and}\quad
\frac{9k}{13}+\frac{6k}{11}=\frac{177k}{(13)(11)}.$$

Now calculate the ratio. There is some nice cancellation. The ratio of X to Y is $109:177$.

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