Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #5
Grade 4 geometry-2dPick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
One sentence hides three short tasks, so Tool #7 (Break into Subproblems) keeps the work in order: (i) find the rectangle's perimeter and area, (ii) reuse that perimeter to get the square's side and area, (iii) subtract to get the increase. Tool #1 (Draw a Diagram) is the visual support — sketching the 60 × 20 rectangle next to a square shows that "same fence" means equal perimeter, which is the only link between the two shapes.
Find the fence and the area
Subproblem 1: the rectangle's perimeter is the reusable fence length 160 ft, and its area is 1200 ft².
Perimeter and area of a rectangle are Grade 3 formulas. The perimeter 160 is the length of fence we get to reuse.
3.MD.D.8Identify SubproblemsFind the square's area
Subproblem 2: the same 160 ft fence makes a square of side 40 ft, so its area is 1600 ft².
P = 4s for a square, so s = P/4. Area of a square is side × side, another Grade 3 standard.
Reshaped from the same fence, the square has a side of 40 ft and covers 1600 square feet.
▸ Why?
The square is built from the very same fence, so its border is the same total length as the rectangle's border, 160 ft.
▸ Why?
The fence is one fixed length of material; bending it into a new shape neither adds nor removes any of it, so going once around either garden uses the whole 160 ft.
▸ Why?
One side of the square is 40 ft: the four equal sides must together use up the whole 160 ft, and sharing 160 into 4 equal parts gives 40.
▸ Why?
Four equal sides that total 160 means each side is one of four equal groups making 160, and recovering that one group undoes the times-4: 160 ÷ 4 = 40.
▸ Why?
A 40-by-40 square is tiled by 40 equal rows with 40 unit squares in each row, so its area is 40 × 40 = 1600 square feet.
▸ Why?
Area counts the unit squares that fill the shape, and 40 equal rows of 40 is the same amount added over and over, which is exactly what multiplying side by side does.
Subtract for the increase
Subproblem 3: subtract the old area from the new area — the garden grows by 400 ft².
"By how many" is a comparison subtraction. The Grade 4 four-operations standard covers exactly this kind of multi-step word-problem finish.
4.OA.A.3Identify SubproblemsSame fence means same perimeter — and for a fixed perimeter a square holds more area than a long, thin rectangle. Three Grade 3-4 steps (perimeter, side, subtract) turn this AMC 8 problem into routine work.
- Find the fence and the area
- Find the square's area
- Subtract for the increase
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