2003 AMC 10B Problem 5

P1-10~2 minPrint

Linear Equations & Word Problems Sets, Estimation & Miscellaneous Set up the equation/formula and compute; no special trick needed

Moe 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?

Show solution without answering

Key insight

Because of the 4-inch overlap, each pass mows only 24 inches = 2 feet of new grass. Walking 5000 feet an hour then clears 10000 square feet an hour.

Solution

1. The mower cuts $28$ inches, but $4$ inches of each pass is grass that was already cut. So each pass mows only $28 - 4 = 24$ inches of new grass. That is exactly $2$ feet.
2. In one hour Moe walks $5000$ feet. So in one hour he mows a strip $5000$ feet long and $2$ feet wide:
$5000 \times 2 = 10000 \text{ square feet per hour}.$
3. The lawn is $90 \times 150 = 13500$ square feet.
4. Time needed:
$13500 \div 10000 = 1.35 \text{ hours}.$

The answer is $\boxed{\textbf{(C)}\ 1.35}$ .

Why this works

Mowing speed in area per hour is walking speed times the useful width of the cut. The overlap only shrinks that useful width. Put everything in the same unit (feet) before you multiply, and the rest is one division.

Alternative approach

Picture the strips running along the $90$ -foot side. Each strip covers $2$ feet of the $150$ -foot side, so Moe needs $150 \div 2 = 75$ strips of $90$ feet. He walks $75 \times 90 = 6750$ feet, which takes $6750 \div 5000 = 1.35$ hours.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Because each pass overlaps the previous one by $4$ inches, every pass adds only $28 - 4 = 24$ inches of new grass, which is exactly $2$ feet.

While walking $5000$ feet in an hour, Moe therefore clears a strip $5000$ feet long and $2$ feet wide: $10000$ square feet per hour.

The lawn has area $90 \cdot 150 = 13500$ square feet, so the time needed is
$\frac{13500}{10000} = 1.35 \text{ hours}.$

The answer is $\boxed{\textbf{(C)}\ 1.35}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.2, §2.3)
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 7: Ratios, Conversions, and Rates (§7.4)