AMC 10 · 2003 · #4

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain
area-rectanglesunit-conversionrate dimensional-analysisidentify-subproblems ↑ Prerequisites: unit-conversionarea-rectangles
📏 Medium solution 💡 2 insights
Problem
Moe mows a rectangular lawn that is 90 feet by 150 feet. His mower cuts a strip 28 inches wide, but he overlaps each pass by 4 inches. He walks 5000 feet per hour. Find which choice is closest to the number of hours the mowing takes.

Pick an answer.

(A)
0.75
(B)
0.8
(C)
1.35
(D)
1.5
(E)
3
How to solve
Strategy Analyze the Units

The question is a chain of rates and measurements — inches, feet, square feet, feet-per-hour — so Tool #8 (Analyze the Units) is the spine: keeping every quantity in feet forces the arithmetic to line up and stops the classic mistake of dividing area straight by speed. Tool #7 (Identify Subproblems) splits the job into three clean pieces: the real cutting width, the lawn's area, and the total walking distance, each a single easy step. Tool #3 (Eliminate Possibilities) then guards the finish: forgetting the overlap or forgetting the swath entirely lands on one of the trap choices, so checking the units confirms the answer is the one that survives.

1STEP 1

Find the real cutting width

The overlap means fresh grass per pass is 28 minus 4, or 2 feet once converted.

28-4=24 in=24/12=2 ft
2STEP 2

Find the lawn's area

The lawn's area is 90 times 150, or 13500 square feet.

90×150=13500 ft²
3STEP 3

Turn the area into walking distance

Seen as one long strip, the walking distance is the area over the width: 6750 feet.

(13500 ft²)/(2 ft)=6750 ft
4STEP 4

Divide distance by speed

Dividing by the walking speed gives 1.35 hours, choice (C).

(6750 ft)/(5000 ft/hr)=1.35 hr → (C)
Answer
1.35
Walking 5000 feet takes exactly one hour, and 6750 feet is a bit more than that but well under 7500 feet (which would be 1.5 hours), so the time should sit between 1 and 1.5 hours — and 1.35 does. The biggest trap is (E) 3: dividing the area 13500 straight by the speed 5000 gives 2.7≈3, but that ignores the swath width and mixes square feet with feet, so the units expose it as wrong. Keeping the 2-foot cutting width in play is what lands the answer on (C).
💡Key takeaway

Turn the whole mowed lawn into one long strip: its area divided by its width is the distance you walk, and distance divided by speed is the time.

  • Find the real cutting width
  • Find the lawn's area
  • Turn the area into walking distance
  • Divide distance by speed

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