AMC 10 · 2016 · #11
Grade 6 geometry-2dPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The post counts on the two sides are linked by a 'twice as many' relationship and a fixed total, so naming the shorter-side count with a variable turns the words into one equation. A quick sketch keeps the shared corner posts from being counted twice, and once a side's post count is known the length is a small multiplication subproblem.
Name the posts per side
Let n be the posts on a shorter side (corners included); a longer side then has twice as many, 2n posts.
Giving the unknown count a name lets the 'twice as many' rule become arithmetic instead of guesswork.
6.EE.B.6Use Matrix LogicCount posts around the rectangle
Summing all four sides gives 2n + 2(2n), but the 4 shared corners are double-counted, so subtract 4; the total is 20.
A corner post is shared by two sides, so counting it once for each side overcounts it by exactly one.
6.EE.B.7Convert To AlgebraSolve for n
Combining, 6n - 4 = 20, so 6n = 24 and n = 4; a shorter side has 4 posts, a longer side 8.
Undoing the subtraction and then the multiplication peels the equation back to the single value of n.
6.EE.B.7Use Matrix LogicTurn posts into side lengths
Length comes from gaps, one fewer than posts: 4 posts → 3 gaps → 12 yards; 8 posts → 7 gaps → 28 yards.
Posts are fenceposts, not fence pieces, so a row of posts has one fewer gap than it has posts.
4.OA.A.3Draw A DiagramCompute the area
The garden is 12 by 28 yards, so its area is 12 × 28 = 336 square yards — choice (B).
Area of a rectangle is just the two side lengths multiplied together.
4.MD.A.3Identify SubproblemsCount the gaps, not the posts: a row of posts always has one fewer space than posts, and corners get shared by two sides.
- Name the posts per side
- Count posts around the rectangle
- Solve for n
- Turn posts into side lengths
- Compute the area