3. Roza has a garden that is divided into rectangular sections as shown in the image (dimensions are marked in meters). In each section, she plans to plant a different type of flower. She will plant begonias, lilies, daisies, roses, and irises, with one plant per square meter of the garden.

She inquired about prices from her neighbor Mirta. Iris bulbs are only sold in packs of 5, while seedlings of other flowers are sold individually. Mirta paid $39 \mathrm{kn}$ for three begonia seedlings, $73 \mathrm{kn}$ for four lily seedlings, $75 \mathrm{kn}$ for two daisy seedlings, $83 \mathrm{kn}$ for two rose seedlings, and $104 \mathrm{kn}$ for a pack of five iris bulbs. What is the minimum amount Roza needs to spend on buying the flowers, ensuring that all purchased seedlings and bulbs are planted in her garden?

![](https://cdn.mathpix.com/cropped/2024_05_30_9fa22177d42fb47260cfg-3.jpg?height=393&width=632&top_left_y=194&top_left_x=1191)
## Solution.

First, we need to determine the areas of each part of the garden.

The dimensions of the unmarked rectangle are $6-4=2 \text{~m}$ and $3-1=2 \text{~m}$.

The areas of the individual sections are marked on the diagram:

| $20 \text{~m}^{2}$ | $21 \text{~m}^{2}$ |  |
| :---: | :---: | :---: |
|  | $4 \text{~m}^{2}$ | $15 \text{~m}^{2}$ |
| $6 \text{~m}^{2}$ |  |  |

Iris bulbs are sold in packs of 5, so Roza must plant them on an area of $15 \text{~m}^{2}$ or $20 \text{~m}^{2}$.

Let's determine the prices of the flowers (per unit) and arrange them from the lowest to the highest.

begonia seedling $\quad 39 \text{kn} : 3 = 13 \text{kn}$

lily seedling $\quad 73 \text{kn} : 4 = 18 \text{kn} \text{ and } 25 \text{lp}$

dahlia seedling $\quad 75 \text{kn} : 2 = 37 \text{kn} \text{ and } 50 \text{lp}$

rose seedling $\quad 83 \text{kn} : 2 = 41 \text{kn} \text{ and } 50 \text{lp}$

iris bulb $\quad 104 \text{kn} : 5 = 20 \text{kn} \text{ and } 80 \text{lp}$

The order of flower varieties by price (per unit) from the lowest to the highest:

begonia seedling, lily seedling, iris bulb, dahlia seedling, rose seedling.

The total cost will be the lowest if cheaper flowers are planted on larger areas, and more expensive flowers on smaller areas. The best choice would be:

begonias on an area of $21 \text{~m}^{2}$

lilies on an area of $20 \text{~m}^{2}$

irises on an area of $15 \text{~m}^{2}$

dahlias on an area of $6 \text{~m}^{2}$

roses on an area of $4 \text{~m}^{2}$

Note that the irises are on an area of $15 \text{~m}^{2}$, so all the bulbs will be used.

|  | area | Mirta | cost for Roza's garden |
| :--- | :--- | :--- | :--- |
| begonias | $21 \text{~m}^{2}$ | 3 for $39 \text{kn}$ | $7 \cdot 39 = 273 \text{kn}$ |
| lilies | $20 \text{~m}^{2}$ | 4 for $73 \text{kn}$ | $5 \cdot 73 = 365 \text{kn}$ |
| irises | $15 \text{~m}^{2}$ | 5 for $104 \text{kn}$ | $3 \cdot 104 = 312 \text{kn}$ |
| dahlias | $6 \text{~m}^{2}$ | 2 for $75 \text{kn}$ | $3 \cdot 75 = 225 \text{kn}$ |
| roses | $4 \text{~m}^{2}$ | 2 for $83 \text{kn}$ | $2 \cdot 83 = 166 \text{kn}$ |

total: $273 + 365 + 312 + 225 + 166 = 1341 \text{kn}$

Roza will spend at least $1341 \text{kn}$.