AMC 8 · 2012 · #9

Grade 4 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

systems-of-equationslinear-equations-two-var convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticlinear-equations-one-var

📏 Medium solution 💡 3 insights

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Problem

At the zoo, Margie counts 200 heads and 522 legs across two-legged birds and four-legged mammals. How many of those 200 animals are birds?

Pick an answer.

(A)

$hspace{.05in}61$

(B)

$hspace{.05in}122$

(C)

$hspace{.05in}139$

(D)

$hspace{.05in}150$

(E)

$hspace{.05in}161$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The reference solution uses algebra (Tool #13), but a 4th grader can crack this without variables. Tool #9 (Easier Related Problem): pretend every animal is a bird and count legs — that gives 400. The real count is 522, so the extra legs must come from swapping some birds for mammals. Each swap adds 2 legs, so dividing the leg shortage by 2 instantly gives the mammal count. Tool #6 (Guess and Check) is a clean backup: plug each answer choice into birds · 2 + (200 - birds) · 4 and see which gives 522.

1STEP 1

Pretend all 200 animals are two-legged birds; then the legs would be just 200 × 2 = 400.

200 × 2 = 400 legs

2STEP 2

Real legs are 522, but the all-bird version has only 400, so the extra 122 legs come from mammals.

522 - 400 = 122 extra legs

3STEP 3

One bird→mammal swap adds 4 - 2 = 2 legs, so 122 ÷ 2 = 61 mammals.

mammals =

\frac{122}{2}

= 61

4STEP 4

Heads stay at 200, so birds = 200 - 61 = 139.

birds = 200 - 61 = 139 → (C)

Answer

hspace{.05in}139

Plug the answer back in: 139 birds give 139 × 2 = 278 legs, 61 mammals give 61 × 4 = 244 legs, and 278 + 244 = 522. Heads: 139 + 61 = 200. Both totals match exactly. Also a sanity check — birds far outnumber mammals because 522 is much closer to 400 (all-bird) than to 800 (all-mammal), so the answer should be near the bird end, and 139 is indeed in that range.

💡Key takeaway

Pretend every animal is a bird first — then each extra pair of legs you're missing is one mammal hiding in the flock. Pure Grade 4 arithmetic, no algebra needed!

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