Competition · AMC preparation · step 4 of 4
AMC 8 · 2012 · #9
Grade 4 algebraPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Assume every animal is a bird
Pretend all 200 animals are two-legged birds; then the legs would be just 200 × 2 = 400.
Replacing the mixed flock with an all-bird flock turns a 2-unknown puzzle into a single multiplication — exactly the Grade 3 "multiply within 100 to solve word problems" move.
3.OA.A.3Solve An Easier Related ProblemCompare with the real leg count
Real legs are 522, but the all-bird version has only 400, so the extra 122 legs come from mammals.
The gap between the easy version and the real version measures exactly how much "mammal-ness" is in the flock — a Grade 4 multi-step word-problem reasoning move.
4.OA.A.3Solve An Easier Related ProblemDivide the extra legs by 2
One bird→mammal swap adds 4 - 2 = 2 legs, so 122 ÷ 2 = 61 mammals.
Each bird-to-mammal swap is worth +2 legs, so dividing the leg surplus by 2 counts the mammals directly — no algebra needed.
The number of four-legged mammals equals the leg surplus of 122 over the all-bird count divided by 2.
▸ Why?
The 122 legs beyond the all-bird total come only from mammals, and every mammal is responsible for the very same number of extra legs.
▸ Why?
Trading one bird for one mammal leaves the single head untouched but changes its legs from 2 to 4, so it adds exactly 2 legs.
▸ Why?
A mammal's 4 legs break with no overlap into the 2 legs a bird already has plus 2 more, so the added part is exactly 2.
▸ Why?
Since the surplus is that same 2 legs counted once for each mammal, the surplus is a run of equal groups of 2, one group per mammal.
▸ Why?
To recover how many groups of 2 build up the 122 surplus, you divide, because division reverses the repeated grouping that multiplication does.
Subtract to get the birds
Heads stay at 200, so birds = 200 - 61 = 139.
Heads are conserved at 200, so once one species is counted, the other is just a subtraction.
4.OA.A.3Solve An Easier Related ProblemPretend 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!
- Assume every animal is a bird
- Compare with the real leg count
- Divide the extra legs by 2
- Subtract to get the birds
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