AMC 10 · 2012 · #7

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

For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A chipmunk puts 3 acorns in each hole it digs; a squirrel puts 4 acorns in each hole it digs. Both animals hid the same total number of acorns, but the squirrel used 4 fewer holes. Find how many acorns the chipmunk hid.

Givens: The chipmunk hides 3 acorns per hole.; The squirrel hides 4 acorns per hole.; Both hid the same total number of acorns.; The squirrel dug 4 fewer holes than the chipmunk.

Unknowns: The number of acorns the chipmunk hid (equivalently, how many holes the chipmunk dug).

Understand

Givens: The chipmunk hides 3 acorns per hole.; The squirrel hides 4 acorns per hole.; Both hid the same total number of acorns.; The squirrel dug 4 fewer holes than the chipmunk.

Plan

Primary tool: #4 Introduce a Variable

Secondary: #13 Convert to Algebra

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.

Execute — Answer: D

#4 Introduce a Variable 6.EE.B.6 Step 1

Name the hole count

Let h be the number of holes the chipmunk dug.
Since the chipmunk puts 3 acorns in each hole, it hid 3h acorns.
The squirrel dug 4 fewer holes, so it dug h - 4 holes, and since it puts 4 acorns in each hole, it hid 4(h - 4) acorns.

$$\text{chipmunk} = 3h, \quad \text{squirrel} = 4(h - 4)$$

💡 One letter for the chipmunk's holes lets both animals' totals be written in the same unknown.

#13 Convert to Algebra 7.EE.B.4 Step 2

Set the totals equal

The problem says both animals hid the same number of acorns.
So the chipmunk's total 3h equals the squirrel's total 4(h - 4).
That gives one equation in one unknown.

$$3h = 4(h - 4)$$

💡 The words 'the same number of acorns' are exactly an equals sign.

#13 Convert to Algebra 7.EE.B.4 Step 3

Solve for the holes

Expand the right side: 4(h - 4) = 4h - 16.
So 3h = 4h - 16.
Subtract 3h from both sides to get 0 = h - 16, which means h = 16.
The chipmunk dug 16 holes.

$$3h = 4h - 16 \;\Rightarrow\; 16 = h$$

💡 Gathering every h on one side leaves the hole count alone.

#4 Introduce a Variable 3.OA.A.1 Step 4

Count the acorns

The chipmunk hid 3 acorns in each of its 16 holes, so it hid 3 x 16 = 48 acorns.
That matches answer choice (D).

$$3 \times 16 = 48$$

💡 Multiply holes by acorns-per-hole to get the total, so the answer is (D).

[1] #4 6.EE.B.6 Let h be the number of holes the chipmunk dug. Since the chipmunk puts 3 acorns

[2] #13 7.EE.B.4 The problem says both animals hid the same number of acorns. So the chipmunk's t

[3] #13 7.EE.B.4 Expand the right side: 4(h - 4) = 4h - 16. So 3h = 4h - 16. Subtract 3h from bot

[4] #4 3.OA.A.1 The chipmunk hid 3 acorns in each of its 16 holes, so it hid 3 x 16 = 48 acorns.

Review

Reasonableness: 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.

Alternative: Without algebra, note the shared total must be a multiple of both 3 and 4, so a multiple of 12. For a total of 12 the holes are 4 and 3 (difference 1); each extra 12 acorns adds one more hole to the gap. A gap of 4 holes needs 4 x 12 = 48 acorns, again giving (D).

CCSS standards used (min grade 7)

6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Writing the two totals 3h and 4(h - 4) from the same unknown hole count.)
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities (Setting 3h = 4(h - 4) and solving the linear equation for h = 16.)
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups (Computing 3 x 16 = 48 acorns from 16 holes of 3 acorns each.)

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

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Name the hole count

Set the totals equal

Solve for the holes

Count the acorns

Review

CCSS standards used (min grade 7)

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