Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #13
Grade 6 arithmeticPick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We are not asked for each box's weight — only the total. Tool #11 (Find an Invariant) says to look for a quantity that stays the same no matter which pair you pick. The key invariant here is the count: when you add all three pair weights, every box gets counted exactly twice. Tool #4 (Introduce a Variable) lets us name the box weights x, y, z so we can write that observation as a clean equation. Then the total x + y + z falls out by dividing by 2 — no need to solve for x, y, z separately.
Name the box weights
Name the three box weights x, y, z; each pair weighing then gives one equation.
Using letters for unknown weights is the Grade 6 move: "use variables to represent numbers and write expressions when solving a real-world problem."
6.EE.B.6Introduce A VariableAdd the three equations
Add all three equations; on the left each of x, y, z appears exactly twice.
The "each box counted twice" pattern is the invariant. It does not depend on which pair weights you got — only on the fact that every pair was weighed.
Adding the three pair weights together counts each box's weight exactly twice.
▸ Why?
Each pair weight is just the two boxes on that weighing added together, so the three pair weights together hold six box weights, and sorting those weights shows every box turns up exactly twice.
▸ Why?
The three pair weights can be poured into one running total and the box weights inside them shuffled and grouped however is convenient, because rearranging the numbers being added never changes their sum.
▸ Why?
Adding the box weights in any order gives the same total, so they can be lined up to put each box next to its other copy.
▸ Why?
The box weights can be grouped in any way without changing the total, so the two copies of each box can be pushed together and counted as a pair.
▸ Why?
Each box was weighed once with each of the other two boxes, and matching each of those two other boxes to the single weighing it shares with this box shows this box sits in exactly two of the pairs.
Combine like terms
Combine like terms: the left becomes 2(x + y + z), and the right adds to 374.
Factoring out the 2 makes the combined weight x + y + z visible as a single block.
6.EE.A.3Work BackwardsDivide by 2
Divide both sides by 2: the combined weight is x + y + z = 187 pounds.
One-step equation: divide both sides by the same nonzero number. The answer is the combined weight in pounds.
6.EE.B.7Introduce A VariableWhen every pair gets weighed, add all the pair totals — each box was counted twice, so half the sum is the answer.
- Name the box weights
- Add the three equations
- Combine like terms
- Divide by 2
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