AMC 10 · 2003 · #5

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 swath 28 inches wide, but he overlaps each pass by 4 inches so that no grass is missed. He walks 5000 feet per hour while pushing the mower. Which choice is closest to the number of hours it takes him to mow the lawn?

Pick an answer.

(A)
0.75
(B)
0.8
(C)
1.35
(D)
1.5
(E)
3

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

A 28-inch swath minus the 4-inch overlap leaves 24 inches of new grass per pass, which is 2 feet.

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

Find the lawn's area

The lawn is a rectangle, so its area is 90×150=13500 square feet of grass to cut.

90×150=13500 ft²
3STEP 3

Turn the area into walking distance

Treat everything mowed as one 2-foot-wide strip: its length is 13500÷2=6750 feet, the distance he walks.

(13500 ft²)/(2 ft)=6750 ft
4STEP 4

Divide distance by speed

Time is distance over speed: 6750÷5000=1.35 hours, which matches choice (C) exactly.

(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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