Competition · AMC preparation · step 4 of 4
AMC 8 · 2012 · #19
Grade 6 algebraPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Each clue is phrased as a complement: "all but 6 are red" tells us T - r = 6, not r directly. Tool #16 (Count the Complement) is exactly that move — read "all but k" as the size of the non-that-color set. Tool #13 (Convert to Algebra) then lets us name T, r, g, b and add the three complement equations. The three left sides add to 3T - (r+g+b) = 3T - T = 2T, and the right sides add to 6 + 8 + 4 = 18, so T falls out in one line.
Rewrite each clue as a complement
Read each clue as a complement: "all but 6 are red" means 6 marbles are not red, so T - r = 6; likewise T - g = 8 and T - b = 4.
"All but k are X" is a complement statement: it counts everything that is NOT X.
6.EE.A.2Change Focus Count The ComplementAdd the three equations
Add the three complement equations. The left side collapses to 3T - (r + g + b) and the right side to 6 + 8 + 4 = 18.
Adding three clean equations packages all three clues into one statement about T.
4.OA.A.3Convert To AlgebraSimplify to find the total
Since r + g + b = T, the left side becomes 3T - T = 2T, so 2T = 18 and T = 9.
Replacing r + g + b with T collapses three unknowns into one, and a one-step linear equation finishes the job.
Adding the three "all but" equations makes the left side collapse to twice the total, so 2T = 6 + 8 + 4.
▸ Why?
Written side by side, the three left sides add to (T-r)+(T-g)+(T-b), and rearranging and regrouping the like terms turns this into 3T-(r+g+b).
▸ Why?
The added terms can be reordered so the three T's sit together and the three subtracted color counts sit together.
▸ Why?
Once reordered, adding the three T's first and the three counts first regroups the sum without changing its value.
▸ Why?
The subtracted piece r+g+b is the whole jar T itself, so 3T-(r+g+b) becomes 3T-T, which is 2T.
▸ Why?
Every marble is red, green, or blue with none left out and none counted twice, so the three color counts add back to the total T.
When a problem says "all but k", it's counting the OPPOSITE — write that down, add the clues, and the total appears in one step.
- Rewrite each clue as a complement
- Add the three equations
- Simplify to find the total
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