AMC 10 · 2003 · #3
Easy mode Grade 5Rose's flower bed is a rectangle split into five smaller rectangles. The figure shows the side lengths, in feet. She plants one flower in every square foot, and she uses a different kind of flower in each of the five parts. Each plant has its own price: asters \textdollar1, begonias \textdollar1.50, cannas \textdollar2, dahlias \textdollar2.50, and Easter lilies \textdollar3. She wants the total cost to be as low as possible. What is the least possible cost, in dollars?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A rectangular flower bed is cut into five rectangular regions whose side lengths are given in the figure. Each region is filled with one type of flower, planted one per square foot. Five flower types cost \$1, \$1.50, \$2, \$2.50, and \$3 per plant. Find the smallest total cost, in dollars, if each type is used in exactly one region.
Givens: The bed splits into five rectangular regions.; The region side lengths (in feet) are: a $6\times1$ strip, a $4\times5$ block, a $7\times3$ block, a $5\times3$ block, and a $2\times2$ block.; One plant per square foot fills each region.; Per-plant prices: asters \$1, begonias \$1.50, cannas \$2, dahlias \$2.50, Easter lilies \$3.; Each of the five flower types goes in exactly one region.; Answer choices: (A) $108$, (B) $115$, (C) $132$, (D) $144$, (E) $156$.
Unknowns: The least possible total cost, in dollars, of filling all five regions.
Understand
Restated: A rectangular flower bed is cut into five rectangular regions whose side lengths are given in the figure. Each region is filled with one type of flower, planted one per square foot. Five flower types cost \$1, \$1.50, \$2, \$2.50, and \$3 per plant. Find the smallest total cost, in dollars, if each type is used in exactly one region.
Givens: The bed splits into five rectangular regions.; The region side lengths (in feet) are: a $6\times1$ strip, a $4\times5$ block, a $7\times3$ block, a $5\times3$ block, and a $2\times2$ block.; One plant per square foot fills each region.; Per-plant prices: asters \$1, begonias \$1.50, cannas \$2, dahlias \$2.50, Easter lilies \$3.; Each of the five flower types goes in exactly one region.; Answer choices: (A) $108$, (B) $115$, (C) $132$, (D) $144$, (E) $156$.
Plan
Primary tool: #14 Extreme Principle
Secondary: #7 Identify Subproblems, #8 Analyze the Units
The total cost is a sum of five products (price $\times$ 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.
Execute — Answer: A
4.MD.A.3 Step 1 Find the five region areas
- Each region is a rectangle, so its area (and its number of plants) is length times width.
- From the figure the five rectangles measure $7\times3$, $4\times5$, $5\times3$, $6\times1$, and $2\times2$.
- Multiply each: $7\times3=21$, $4\times5=20$, $5\times3=15$, $6\times1=6$, $2\times2=4$.
💡 A rectangle's area is just its two side lengths multiplied together.
4.OA.A.2 Step 2 Cheapest flower on the biggest region
- The five prices, from low to high, are \$1, \$1.50, \$2, \$2.50, \$3. The five areas, from big to small, are $21$, $20$, $15$, $6$, $4$.
- Because every plant in a region costs the same, a bigger region multiplies its price more times.
- To keep the total small, let the cheapest price hit the largest area and the priciest hit the smallest area — pair the lists in opposite order.
💡 A high price hurts most where it is multiplied by the most plants, so keep high prices on small patches.
5.NBT.B.7 Step 3 Cost of each region
- Multiply each paired price by its area.
- $\$1\times21=\$21$; $\$1.50\times20=\$30$; $\$2\times15=\$30$; $\$2.50\times6=\$15$; $\$3\times4=\$12$.
- Watch the decimals: $1.50\times20$ is $30$, not $3$, and $2.50\times6$ is $15$.
💡 Each region's bill is one price copied once for every square foot it covers.
5.NBT.B.7 Step 4 Add up the least total
- Add the five region costs: $21+30+30+15+12=108$.
- Any other matching puts a higher price on a bigger area somewhere, which can only raise the sum, so $108$ is the smallest possible.
- That is choice (A).
💡 Summing the smartest pairing gives the lowest bill the garden can have.
4.MD.A.3 Each region is a rectangle, so its area (and its number of plants) is length tim 4.OA.A.2 The five prices, from low to high, are \$1, \$1.50, \$2, \$2.50, \$3. The five a 5.NBT.B.7 Multiply each paired price by its area. $\$1\times21=\$21$; $\$1.50\times20=\$30 5.NBT.B.7 Add the five region costs: $21+30+30+15+12=108$. Any other matching puts a highe Review
Reasonableness: The whole bed holds $21+20+15+6+4=66$ plants. If every plant were the cheapest \$1 the bill would be \$66; if every plant were the priciest \$3 it would be \$198. The true answer must sit between, and \$108 does. It also should beat the average-price estimate: the mean price is \$2, giving $2\times66=\$132$, and a smart pairing must come in under that average — \$108 does, while the trap choice (C) $132$ is exactly the do-nothing average and (D) $144$/(E) $156$ are worse than average.
Alternative: Start from the average-cost total $\$2\times66=\$132$ and track the savings from smart pairing. Cheap flowers on big regions save money; expensive on small regions also saves. Compared to charging every region \$2, the discounts and surcharges net to $-\$24$ (e.g. the \$1 flower on $21$ saves $\$1\times21=\$21$, the \$3 flower on $4$ adds $\$1\times4=\$4$, and so on), landing back at $132-24=108$.
CCSS standards used (min grade 5)
4.MD.A.3Apply area and perimeter formulas for rectangles in real-world problems (Finding each region's area (and plant count) as length times width: $21,20,15,6,4$ square feet.)4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison (Reasoning that a fixed price multiplied over a larger area costs more, so the cheapest price should sit on the largest region.)5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Computing each region's dollar cost with decimal prices ($1.50\times20$, $2.50\times6$) and summing to $108$.)
⭐ To spend the least, put the cheapest flower on the biggest patch and the priciest flower on the smallest patch.
⭐ To spend the least, put the cheapest flower on the biggest patch and the priciest flower on the smallest patch.
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