AMC 10 · 2003 · #5
Grade 6 rate-ratioMoe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. He walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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, so the fresh grass he cuts each pass is narrower than 28 inches. He walks 5000 feet per hour. Find which choice is closest to the number of hours the mowing takes.
Givens: Lawn is a rectangle $90$ ft by $150$ ft; Mower swath is $28$ inches wide; Each pass overlaps the previous one by $4$ inches; Moe walks $5000$ feet per hour; Answer choices: (A) $0.75$, (B) $0.8$, (C) $1.35$, (D) $1.5$, (E) $3$
Unknowns: The number of hours it takes Moe to mow the whole lawn
Understand
Restated: 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, so the fresh grass he cuts each pass is narrower than 28 inches. He walks 5000 feet per hour. Find which choice is closest to the number of hours the mowing takes.
Givens: Lawn is a rectangle $90$ ft by $150$ ft; Mower swath is $28$ inches wide; Each pass overlaps the previous one by $4$ inches; Moe walks $5000$ feet per hour; Answer choices: (A) $0.75$, (B) $0.8$, (C) $1.35$, (D) $1.5$, (E) $3$
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
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.
Execute — Answer: C
5.MD.A.1 Step 1 Find the real cutting width
- Each pass cuts a $28$-inch swath, but $4$ inches of it just re-cuts the previous pass.
- So the new grass cut per pass is $28-4=24$ inches.
- Convert to feet: $24$ inches is $\frac{24}{12}=2$ feet, since $12$ inches make a foot.
💡 Overlap means part of each pass is wasted, so the width that counts is the swath minus the overlap.
4.MD.A.3 Step 2 Find the lawn's area
- The lawn is a rectangle, so its area is length times width: $90\times150=13500$ square feet.
- This is the total amount of grass that must be cut.
💡 All the grass has to be covered, and for a rectangle that total is just its two sides multiplied.
4.MD.A.3 Step 3 Turn the area into walking distance
- Think of everything Moe cuts as one long strip $2$ feet wide.
- Its area is $13500$ square feet, so its length — the distance he walks — is the area divided by the width: $13500\div2=6750$ feet.
💡 A strip of known area and known width has a length you get by dividing area by width.
6.RP.A.3 Step 4 Divide distance by speed
- Moe walks $5000$ feet each hour, so the time is the distance divided by the speed: $6750\div5000=1.35$ hours.
- This matches choice (C) exactly, so the answer is (C).
💡 At a steady walking speed, time is simply how far you go divided by how fast you go.
5.MD.A.1 Each pass cuts a $28$-inch swath, but $4$ inches of it just re-cuts the previous 4.MD.A.3 The lawn is a rectangle, so its area is length times width: $90\times150=13500$ 4.MD.A.3 Think of everything Moe cuts as one long strip $2$ feet wide. Its area is $13500 6.RP.A.3 Moe walks $5000$ feet each hour, so the time is the distance divided by the spee Review
Reasonableness: 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 $2.7\approx3$, 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).
Alternative: Tool #7 (Identify Subproblems) by counting strips: mow back and forth in strips parallel to the $90$-foot side, so each strip is $90$ feet long. Each strip covers a $2$-foot width, so the number of strips is $150\div2=75$. The total walking distance is $75\times90=6750$ feet, and dividing by $5000$ feet per hour gives $1.35$ hours — the same (C) without ever computing an area.
CCSS standards used (min grade 6)
5.MD.A.1Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step real world problems (Subtracting the overlap and converting the $24$-inch cutting width into $2$ feet.)4.MD.A.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems (Computing the lawn's area $90\times150=13500$ ft$^2$ and then dividing that area by the $2$-foot width to get the $6750$-foot walking distance.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Using the $5000$ feet-per-hour rate to turn $6750$ feet of walking into $1.35$ hours.)
⭐ 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.
⭐ 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.
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