# 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

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In the end this is just a question how to add fractions: Which part of solid A consists of element X? 4 of 4+9 parts, in other words: $\frac{4}{13}$. Similarly $\frac{5}{5+6}=\frac{5}{11}$ of solid B consist of element X. For solid C we have $$\frac{1}{2}(\frac{4}{13}+\frac{5}{11})=\frac{109}{286}.$$

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answer is 109:177 –  Jay May 13 '12 at 18:16
@jay Simon Markett showed you how many of X is in the total. Thus Y would be 286-109=177, thus you get a ratio of 109:177. Make sure you read the answer carefully, before just posting a snide comment. –  yiyi May 13 '12 at 18:25

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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