2012 AMC 10B Problem 7

P1-10~2 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Both animals hid the same total. Write the chipmunk's total as 3 times its holes and the squirrel's as 4 times 4 fewer holes, and set them equal.

Solution

1. Let the chipmunk dig $h$ holes. With $3$ acorns per hole, it hid $3h$ acorns.
2. The squirrel dug $4$ fewer holes, so it dug $h - 4$ holes. With $4$ acorns per hole, it hid $4(h - 4) = 4h - 16$ acorns.
3. They hid the same number of acorns, so
$3h = 4h - 16.$
4. Take $3h$ away from both sides: $0 = h - 16$ , so $h = 16$ .
5. The question asks for acorns, not holes. The chipmunk hid $3 \times 16 = 48$ acorns.
6. Check: the squirrel dug $16 - 4 = 12$ holes and hid $4 \times 12 = 48$ acorns. The totals match.

The answer is $\boxed{\textbf{(D)}\ 48}$ .

Why this works

"Same total, different amounts per group" gives an equation: amount per group times number of groups, on each side. Another way to see it: the squirrel puts $1$ more acorn in each of its holes than the chipmunk does, and that has to make up for the $4 \times 4 = 16$ acorns in the $4$ holes it skipped. Always reread the question at the end, since here the easy mistake is to answer with the number of holes.

Alternative approach

The total is a multiple of $3$ (chipmunk) and of $4$ (squirrel), so it is a multiple of $12$ . Among the choices, only $36$ and $48$ are.

$36$ acorns: the chipmunk digs $12$ holes and the squirrel digs $9$ . The difference is $3$ , not $4$ .
$48$ acorns: the chipmunk digs $16$ holes and the squirrel digs $12$ . The difference is $4$ . This one works.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Let the chipmunk dig $h$ holes. It hides $3h$ acorns. The squirrel digs $h - 4$ holes with $4$ acorns in each, hiding $4(h - 4)$ acorns. The totals are equal:
$3h = 4(h - 4) = 4h - 16 \quad\Longrightarrow\quad h = 16.$
The chipmunk hid $3 \cdot 16 = 48$ acorns. Check: the squirrel used $12$ holes, and $4 \cdot 12 = 48$ .

The answer is $\boxed{\textbf{(D)}\ 48}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)