AMC 8 · 2012 · #19

Grade 6 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

systems-of-equationslinear-equations-two-varlogical-deduction convert-to-algebraidentify-subproblems ↑ Prerequisites: linear-equations-one-varmulti-digit-arithmetic

📏 Medium solution 💡 3 insights

Problem

A jar holds only red, green, and blue marbles. Three clues are given: "all but 6 are red", "all but 8 are green", and "all but 4 are blue." Find the total number of marbles in the jar.

Pick an answer.

(A)

$hspace{.05in}6$

(B)

$hspace{.05in}8$

(C)

$hspace{.05in}9$

(D)

$hspace{.05in}10$

(E)

$hspace{.05in}12$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

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.

1STEP 1

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.

T - r = 6, T - g = 8, T - b = 4

2STEP 2

Add the three complement equations. The left side collapses to 3T - (r + g + b) and the right side to 6 + 8 + 4 = 18.

(T - r) + (T - g) + (T - b) = 6 + 8 + 4

3STEP 3

Since r + g + b = T, the left side becomes 3T - T = 2T, so 2T = 18 and T = 9.

3T - T = 18 → 2T = 18 → T = 9 → (C)

Answer

hspace{.05in}9

Recover each color count: r = T - 6 = 3, g = T - 8 = 1, b = T - 4 = 5. Total: 3 + 1 + 5 = 9. Check the clues: non-red = 1 + 5 = 6 ✓, non-green = 3 + 5 = 8 ✓, non-blue = 3 + 1 = 4 ✓. All three statements hold, and every count is a non-negative integer, so T = 9 is consistent.

💡Key takeaway

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.

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