AMC 10 · 2012 · #7
Grade 7 arithmeticPick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Name the hole count
Let h be the chipmunk's hole count: it hid 3h acorns, while the squirrel dug h - 4 holes and hid 4(h - 4) acorns.
One letter for the chipmunk's holes lets both animals' totals be written in the same unknown.
6.EE.B.6Introduce A VariableSet the totals equal
Both totals are the same, so 3h = 4(h - 4) — one equation in one unknown.
The words 'the same number of acorns' are exactly an equals sign.
The words the same number of acorns are exactly an equals sign.
▸ Why?
Two expressions naming the same quantity name the same value, so they can be set equal.
▸ Why?
Each animal's total is a count of holes times acorns per hole, so both sides are easy to write.
Solve for the holes
Expand to 3h = 4h - 16, then subtract 3h from both sides: the chipmunk dug h = 16 holes.
Gathering every h on one side leaves the hole count alone.
7.EE.B.4Convert To AlgebraCount the acorns
Each of the 16 holes holds 3 acorns, so the chipmunk hid 3 × 16 = 48 acorns — choice (D).
Multiply holes by acorns-per-hole to get the total, so the answer is (D).
3.OA.A.1Introduce A VariableWhen 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