AMC 10 · 2003 · #4
Grade 5 geometry-2d
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The total cost is a sum of five products (price × area). To make that sum as small as possible you must decide which price lands on which area, so this is a min/max matching problem — Tool #14 (Extreme Principle). The rule it gives is simple: put the cheapest flower on the biggest region and the most expensive flower on the smallest region. Before you can match, Tool #7 (Identify Subproblems) breaks the bed into its five rectangles so each area can be found on its own, and Tool #8 (Analyze the Units) keeps the bookkeeping honest — square feet become number of plants, and plants times dollars-per-plant give dollars.
Find the five region areas
Area = length × width, so 7×3, 4×5, 5×3, 6×1, 2×2 give 21, 20, 15, 6, 4 square feet of plants.
A rectangle's area is just its two side lengths multiplied together.
4.MD.A.3Identify SubproblemsCheapest flower on the biggest region
Pair prices $1, $1.50, $2, $2.50, $3 with areas 21, 20, 15, 6, 4 in opposite order — cheapest flower on the largest region.
A high price hurts most where it is multiplied by the most plants, so keep high prices on small patches.
A high price hurts most where it is multiplied by the most plants, so keep the high prices on the small patches.
▸ Why?
Each region's bill is one price repeated once per plant, which is exactly what multiplying records.
▸ Why?
Swapping a higher price onto a bigger region can only raise the total, so the pairing is forced.
Cost of each region
Multiply each pair: 1×21=21, 1.50×20=30, 2×15=30, 2.50×6=15, 3×4=12 dollars — note 1.50×20 is 30, not 3.
Each region's bill is one price copied once for every square foot it covers.
5.NBT.B.7Analyze The UnitsAdd up the least total
Adding gives 21+30+30+15+12=108; any other pairing puts a costlier flower on a bigger area, so 108 is least — choice (A).
Summing the smartest pairing gives the lowest bill the garden can have.
5.NBT.B.7Extreme PrincipleTo spend the least, put the cheapest flower on the biggest patch and the priciest flower on the smallest patch.
- Find the five region areas
- Cheapest flower on the biggest region
- Cost of each region
- Add up the least total