Track metal amounts through three alloys
A piece of alloy made from gold, silver and bronze weighs 136 g. A second piece of alloy weighs 323 g and contains the same amount of gold, twice as much silver, and three times as much bronze as the first piece. A third piece of alloy weighs 238 g and contains 4 times as much gold, one third as much silver, and same amount of bronze as the second piece. Find the respective amounts of gold, silver and bronze used in the first piece of alloy. marks
[2009/RJC/Promo/1]
Solution
Forming linear systems
A piece of alloy made from gold, silver and bronze weighs 136 g. A second piece of alloy weighs 323 g and contains the same amount of gold, twice as much silver, and three times as much bronze as the first piece. A third piece of alloy weighs 238 g and contains 4 times as much gold, one third as much silver, and the same amount of bronze as the second piece. Find the respective amounts of gold, silver and bronze used in the first piece of alloy.
Hint
Name the unknowns: the gold, silver and bronze in the first piece, since every other piece is described relative to it.
Write the three weights as three equations. Be careful with “as much as the second piece”: it refers to the second piece, not the first.
Solve the system (graphing calculator, or elimination) and interpret the answer in grams.
That is , not . The third piece is described relative to the second.
“One third as much silver” as the second piece is , not .
The answer is amounts in grams; say which metal each number belongs to.
Answer · Q
Gold 5 g, silver 75 g, bronze 56 g.
Define the unknowns
The second and third pieces are described using multiples of these amounts, so they are the only natural unknowns.
First piece
Its total weight is 136 g.
Second piece
Same gold (), twice the silver () and three times the bronze () as the first piece.
Third piece
It has 4 times as much gold as the second piece (), one third as much silver () and the same bronze ().
Solve
For the calculator used in the source, open Apps → PlySmlt2 → Simultaneous Eqn Solver, choose 3 equations and 3 unknowns, enter each row of coefficients and its total, then Solve. Alternatively, by elimination: (2) − (1) gives ; (3) − 4(1) gives , i.e. . Substituting gives , then and .
Check in all three equations
Each equation is satisfied, so the point lies on all three planes.
Explore the idea