Stephanie, Mike , and Howard gather blackberries for their grandmother. Stephanie picks 4 3/8 pounds, Mike picks 2.5 pounds, and
2 answers:
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Let and
be the amounts of available alloy (AA) and pure magic steel that are used, so we also have
DA13 is supposed to contain 7% gold, 3% silver, and 90% steel, so that 2.7 kg of it is made up of
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; kg of it will contain
kg of gold,
kg of silver, and
kg of steel. Each kg of magic steel of course contributes 1 kg of steel;
kg of it will contribute
kg of steel.
Then the dwarves need
• total gold:
• total silver:
• total steel:
Solve for and
. The first two equations are consistent and give
, and substituting this into the third we find
. So the dwarves must combine 0.90 kg of AA and 1.80 kg of magic steel.