AMC 8 · 2000 · #16

Grade 4 geometry-2darithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
area-rectanglesperimetersystems-of-equationslinear-equations-two-var convert-to-algebraidentify-subproblems ↑ Prerequisites: area-rectanglesperimeter
📏 Medium solution 💡 3 insights
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Problem
Mateen's rectangular backyard has the property that walking its length 25 times covers 1000 m, and walking its perimeter 10 times also covers 1000 m. Find the area of the backyard in square meters.

Pick an answer.

(A)
40
(B)
200
(C)
400
(D)
500
(E)
1000

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

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.

1STEP 1

Translate each sentence into an equation: 25L = 1000 for the length, and 10P = 1000 for the perimeter P.

25L = 1000, 10P = 1000
2STEP 2

Solve for the length: divide 25L = 1000 by 25 to get L = 40 m.

L = 100025\frac{1000}{25} = 40 m
3STEP 3

Solve for the perimeter: divide 10P = 1000 by 10 to get P = 100 m.

P = 100010\frac{1000}{10} = 100 m
4STEP 4

Put P = 100 and L = 40 into P = 2L + 2W and solve for the width: W = 10 m.

100 = 2(40) + 2W → 100 - 80 = 2W → W = 202\frac{20}{2} = 10 m
5STEP 5

Multiply length by width: A = 40 × 10 = 400 m², choice (C).

A = 40 × 10 = 400 m² → (C)
Answer
400
Check the dimensions back against the original sentences. Length L = 40 m, so 25 lengths give 25 × 40 = 1000 m — matches. Perimeter P = 2(40) + 2(10) = 80 + 20 = 100 m, so 10 perimeters give 10 × 100 = 1000 m — also matches. The area 400 m² falls between trap (A) 40 (the length alone, forgetting to multiply by width) and (D) 500 or (E) 1000 (which mix area with perimeter or total distance). Answer (C) 400 is the only choice consistent with 40 × 10.
💡Key takeaway

Each 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).

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