AMC 10 · 2012 · #3

Grade 7 arithmetic
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
linear-equations-one-varsystems-of-equations convert-to-algebra ↑ Prerequisites: linear-equations-one-var
📏 Medium solution 💡 2 insights
Problem
Two animals hide equal totals, one packing more per hole but using fewer holes. Find one animal's total.

Pick an answer.

(A)
30
(B)
36
(C)
42
(D)
48
(E)
54
How to solve
Strategy Introduce a Variable

The two animals share one hidden quantity, the total number of acorns. Naming the chipmunk's hole count with a single letter lets both totals be written in that same letter, and the phrase 'same number of acorns' turns directly into an equation to solve.

1STEP 1

Name the hole count

One letter names both hole counts.

chipmunk = 3h, squirrel = 4(h - 4)
2STEP 2

Set the totals equal

The equal totals give one equation.

3h = 4(h - 4)
3STEP 3

Solve for the holes

Solving gives 16 holes.

3h = 4h - 16 → 16 = h
4STEP 4

Count the acorns

The total hidden is 48, choice (D).

3 × 16 = 48
Answer
48
Check the two totals with h = 16. The chipmunk: 3 x 16 = 48 acorns in 16 holes. The squirrel: 16 - 4 = 12 holes, 4 x 12 = 48 acorns. Both totals are 48 and the squirrel used 4 fewer holes, so every condition holds.
💡Key takeaway

When two things are equal, call the mystery number one letter, write both sides with it, and let the equation find it.

  • Name the hole count
  • Set the totals equal
  • Solve for the holes
  • Count the acorns

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