2012 AMC 10B Problem 7

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 $\boxed{\textbf{(D)}\ 48}$.

"Same total, different rates" is a one-variable linear equation: rate times count on each side. The squirrel's higher rate saves exactly $4$ holes, so $4$ holes of the squirrel's $4$-acorn rate must equal the chipmunk's extra $1$ acorn per hole over all $h$ holes: $16 = h$. Set up the equation, solve, then remember to answer the question actually asked.

The answer is a multiple of $3$ and of $4$, so it is a multiple of $12$: only $36$ and $48$ qualify. For $36$: $12$ chipmunk holes versus $9$ squirrel holes, a difference of $3$, not $4$. For $48$: $16$ versus $12$, difference $4$. So $48$.

• Solving for the number of holes (16) and reporting it, or writing the squirrel's holes as h + 4 instead of h - 4.
• Setting $3h = 4h - 4$ (subtracting holes instead of acorns), which gives $h = 4$ and a total of $12$, not among the choices.