Percentage question

Question

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

Answer 1

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.

Answer 2

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

How much copper do the 100kg of alloy contain?

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

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

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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.