AMC 10 · 2009 · #8
Grade 8 algebraPick an answer.
Two hidden quantities never change: the bucket's own weight and the weight of a full load of water. Name each with a variable, turn the two 'how full' facts into two equations, and the full-bucket weight falls out by combining them.
Name the two hidden weights
Two hidden weights never change.
Give the unchanging parts names so you can track them through every case.
6.EE.B.6Introduce A VariableWrite each fullness as an equation
Each fill level becomes one equation.
A fraction of the full water plus the same bucket gives each measured weight.
7.EE.B.4Convert To AlgebraSubtract to isolate the water
Subtracting isolates the contents.
Lining up the two equations lets the shared bucket weight erase itself.
Lining up the two equations lets the shared bucket weight erase itself.
▸ Why?
Both weights carry the same bucket, so subtracting removes it entirely.
▸ Why?
Subtracting one true equation from another keeps the result true, so the elimination is legitimate.
Recover the bucket weight
Back-substituting recovers the container.
Once the water is known, one equation reveals the bucket.
8.EE.C.7Convert To AlgebraAdd bucket plus full water
Adding both gives 3a - 2b, choice (D).
Full weight is just the bucket plus one whole load of water.
7.EE.A.1Introduce A VariableName the parts that never change, write one equation per clue, then subtract to make the part you don't want disappear.
- Name the two hidden weights
- Write each fullness as an equation
- Subtract to isolate the water
- Recover the bucket weight
- Add bucket plus full water