Question

To fight a fire breathing dragon, a knight needs a sword made of Dragon Alloy #13, which contains 7% gold, 3% silver, and 90% magic steel. The dwarves promised to forge such a sword for the knight. At the moment, they have an alloy that contains magic steel, 21% gold, and 9% silver. How much of that alloy and how much magic steel do the dwarves have to combine to get 2.7 kg of Dragon Alloy #13?

Answer 1

Let $x$ and $y$ be the amounts of available alloy (AA) and pure magic steel that are used, so we also have

$x + y = 2.7$

DA13 is supposed to contain 7% gold, 3% silver, and 90% steel, so that 2.7 kg of it is made up of

$0.07 \times 2.7 \,\mathrm{kg} = 0.189 \,\mathrm{kg} \text{ gold}$

$0.03 \times 2.7 \,\mathrm{kg} = 0.081 \,\mathrm{kg} \text{ silver}$

$0.90 \times 2.7 \,\mathrm{kg} = 2.43 \,\mathrm{kg} \text{ magic steel}$

For each kg of the available alloy (AA), there is a contribution of 0.21 kg of gold, 0.09 kg of silver, and therefore 0.70 kg of steel; $x$ kg of it will contain $0.21x$ kg of gold, $0.09x$ kg of silver, and $0.70x$ kg of steel. Each kg of magic steel of course contributes 1 kg of steel; $y$ kg of it will contribute $y$ kg of steel.

Then the dwarves need

• total gold: $0.21x = 0.189$

• total silver: $0.09x = 0.081$

• total steel: $0.70x + y = 2.43$

Solve for $x$ and $y$. The first two equations are consistent and give $x = 0.90$, and substituting this into the third we find $y = 1.80$. So the dwarves must combine 0.90 kg of AA and 1.80 kg of magic steel.