AMC 10 · 2003 · #4

Grade 5 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

area-rectanglesoptimizationdecimal-arithmetic optimization-counting ↑ Prerequisites: area-rectangles

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ $1$ each, begonias$ $1.50$ each, cannas $ $2$ each, dahlias$ $2.50$ each, and Easter lilies $$$3$ each. What is the least possible cost, in dollars, for her garden?

Pick an answer.

(A)

108

(B)

115

(C)

132

(D)

144

(E)

156

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.

View mode:

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

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

#7 Identify Subproblems 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: $7\times3=21$, $4\times5=20$, $5\times3=15$, $6\times1=6$, $2\times2=4$.

$21,\;20,\;15,\;6,\;4$ square feet

💡 A rectangle's area is just its two side lengths multiplied together.

#14 Extreme Principle 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.

$\$1\leftrightarrow21,\;\$1.50\leftrightarrow20,\;\$2\leftrightarrow15,\;\$2.50\leftrightarrow6,\;\$3\leftrightarrow4$

💡 A high price hurts most where it is multiplied by the most plants, so keep high prices on small patches.

#8 Analyze the Units 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 $15$.

$21,\;30,\;30,\;15,\;12$ dollars

💡 Each region's bill is one price copied once for every square foot it covers.

#14 Extreme Principle 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).

$$21+30+30+15+12=108\;\Rightarrow\;\textbf{(A)}$$

💡 Summing the smartest pairing gives the lowest bill the garden can have.

[1] #7 4.MD.A.3 Each region is a rectangle, so its area (and its number of plants) is length tim

[2] #14 4.OA.A.2 The five prices, from low to high, are \$1, \$1.50, \$2, \$2.50, \$3. The five a

[3] #8 5.NBT.B.7 Multiply each paired price by its area. $\$1\times21=\$21$; $\$1.50\times20=\$30

[4] #14 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.3 Apply 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.2 Multiply 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.7 Add, 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Find the five region areas

Cheapest flower on the biggest region

Cost of each region

Add up the least total

Review

CCSS standards used (min grade 5)

More like this