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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?

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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 * 2.7 \,\mathrm{kg} = 0.189 \,\mathrm{kg} \text{ gold}


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


0.90 * 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.

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