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Certain alloy contains 20% of copper and 5% of tin. How much of copper and how much tin must be added to 100kg of this alloy to get another with 30% of copper and 10% of tin?

I can't figure how to start it

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How much copper do the 100kg of alloy contain? –  Phira May 19 '11 at 15:34
    
20kg of copper. –  Kaeser May 19 '11 at 15:36
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Now, if you add x kg of copper and y kg of tin, how much copper does the result contain? –  Phira May 19 '11 at 15:42
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So, what is the percentage of copper in the result? This should give you one equation. You get a similar equation for the tin. Then you solve the system of two equations. –  Phira May 19 '11 at 15:54
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@user9325: I think your hints are well chosen and add up to an answer. Please put them in an answer box so we can upvote and this question won't come back due to the robot. Then if you want to add more you can edit it. –  Ross Millikan May 19 '11 at 15:57

2 Answers 2

up vote 3 down vote accepted

Given your problem statement, you have the following quantities: 20kg of copper, 5kg of tin and 75kg of the base metal, 100kg total.

You want to add an unknown amount $x$ to the copper such that the new percentage of copper in the alloy is 30%, thus $(20 + x) / (100 + x + y) = 0.3$. Likewise, you are looking for an amount $y$ to add to the amount of tin such that the new percent is 10%, thus $(5 + y) / (100 + x + y) = 0.1$. Likewise, you known that the base percentage is now going to be 60%, thus $75 / (100 + x + y) = 0.6$.

With some algebra you get:

$$20 + x = 0.3 (100 + x + y)$$ $$20 + x = 30 + 0.3x + 0.3y$$ $$-10 = -0.7x + 0.3y$$

$$5 + y = 0.1 (100 + x + y)$$ $$5 + y = 10 + 0.1x + 0.1y$$ $$-5 = 0.1x - 0.9y$$

Which will give you the following system:

$$\pmatrix{ -10 \\ -5 } = \pmatrix{ -0.7 & 0.3 \\ 0.1 & -0.9} \pmatrix { x \\ y }$$

Solving it will yield:

$$\pmatrix{x \\ y } = \pmatrix{ 17.5 \\ 7.5 }$$

Thus, you would need to add 17.5kg of copper and 7.5kg of tin to get a new alloy with 30% copper, 10% tin and 60% base.

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Please check the $y$ solution. I computed $y=7.5$. –  Américo Tavares May 19 '11 at 16:18
    
Thanks for catching the typo, I mistakenly put down 12.5 (5 + 7.5). –  lewellen May 19 '11 at 16:29

My list of leading questions as used in the comment section to the question:

How much copper do the 100kg of alloy contain?

.

Now, if you add x kg of copper and y kg of tin, how much copper does the result contain?

.

If you add x kg of copper and y kg of tin, how much kg does the result have?

.

So, what is the percentage of copper in the result? This should give you one equation. You get a similar equation for the tin. Then you solve the system of two equations.

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1  
+1, for the hints and for the "invisibles". –  Américo Tavares May 19 '11 at 16:09

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