Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #16
Grade 4 geometry-2darithmeticPick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Each English sentence in the problem turns directly into an equation, so Tool #13 (Convert to Algebra) is the natural fit: "25 lengths make 1000 m" becomes 25L = 1000, and "10 perimeters make 1000 m" becomes 10P = 1000. Tool #7 (Identify Subproblems) keeps the work orderly — solve for L first, then for P, then back out W from the perimeter formula, and finally compute the area. Each subproblem is a one-step Grade 4 division or substitution.
Turn the sentences into equations
Translate each sentence into an equation: 25L = 1000 for the length, and 10P = 1000 for the perimeter P.
"n trips of distance d cover D" always means n · d = D — the Grade 4 multiplication-as-equal-groups idea.
4.OA.A.3Convert To AlgebraSolve for the length
Solve for the length: divide 25L = 1000 by 25 to get L = 40 m.
25 × 40 = 1000 is a familiar fact — four quarters make a dollar, so forty quarters make ten dollars.
4.OA.A.3Identify SubproblemsSolve for the perimeter
Solve for the perimeter: divide 10P = 1000 by 10 to get P = 100 m.
Dividing by 10 shifts the digits one place — 1000 → 100.
4.OA.A.3Identify SubproblemsSolve for the width
Put P = 100 and L = 40 into P = 2L + 2W and solve for the width: W = 10 m.
Once two sides (L = 40) of the four-sided trip are accounted for, the other two sides must add to 100 - 80 = 20, so each is 10.
The backyard's width is 10 m, because the perimeter P = 100 m and the length L = 40 m fit the rectangle relation 100 = 2(40) + 2W, which leaves 2W = 20 and so W = 10.
▸ Why?
Walking once around the rectangle traces its four sides in a loop — two long sides of length L and two short sides of width W — so the whole perimeter is those four side-pieces added together, which is L + L + W + W = 2L + 2W.
▸ Why?
The trip around is the whole boundary cut into its four side-pieces with no side skipped and none walked twice, so the pieces add back to the full perimeter.
▸ Why?
Two sides each of length L is the same length added twice, and adding two equal groups of L is the multiplication 2 × L; the two widths give 2 × W the same way.
▸ Why?
With L = 40 the two lengths already use 2 × 40 = 80 m, so the relation reads 100 = 80 + 2W; taking the 80 back out leaves 2W = 20, and undoing the doubling gives W = 10.
Multiply for the area
Multiply length by width: A = 40 × 10 = 400 m², choice (C).
Area of a rectangle is just length times width — the final Grade 4 step.
4.MD.A.3Identify SubproblemsEach sentence in a word problem can become one short equation. Solve them in order — length, then perimeter, then width, then area — and the answer is 40 × 10 = 400 m², choice (C).
- Turn the sentences into equations
- Solve for the length
- Solve for the perimeter
- Solve for the width
- Multiply for the area
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2012 #9 Gr 4Similar levelSame sub-type: Fraction / Expression Sum EvaluationSame techniqueConvert to Algebra Identify Subproblems
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie…
- AMC 8 2002 #9 Gr 4Similar level
Same sub-type: Fraction / Expression Sum EvaluationSame techniqueIdentify SubproblemsProblems 8,9 and 10 use the data found in the accompanying paragraph and table: Juan organizes the stamps in his collect…
- AMC 8 2005 #10 Gr 4Similar levelSame sub-type: Fraction / Expression Sum EvaluationSame techniqueIdentify Subproblems
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3…