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 * 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.