In front of our school, there is a flower bed. One fifth of all the flowers are tulips, two ninths are daffodils, four fifteenths are hyacinths, and the rest are pansies. How many flowers are there in total in the bed if there are no more than 60 and no fewer than 30 of any type? (M. Petrová)

Hint. Could there be, for example, 100 flowers in the bed?
The total number of flowers must be divisible by five, because one fifth of them are tulips. For a similar reason, the total number of all flowers must be divisible by nine (due to daffodils) and fifteen (due to hyacinths). Therefore, the total number of flowers in the bed is some common multiple of the numbers 5, 9, and 15. The least common multiple of these three numbers is 45, so the total number of flowers is a multiple of 45.

In the following table, we go through the multiples of 45 and determine the corresponding numbers of individual types of flowers. We are looking for a row where all these numbers are in the range from 30 to 60 (inclusive).

| total  $(c)$ | tulips  $\left(t=\frac{1}{5} c\right)$ | daffodils  $\left(n=\frac{2}{9} c\right)$ | hyacinths  $\left(h=\frac{4}{15} c\right)$ | primroses  $(m=c-t-n-h)$ |
| :---: | :---: | :---: | :---: | :---: |
| 45 | 9 | 10 | 12 | 14 |
| 90 | 18 | 20 | 24 | 28 |
| 135 | 27 | 30 | 36 | 42 |
| $\mathbf{1 8 0}$ | $\mathbf{3 6}$ | $\mathbf{4 0}$ | $\mathbf{4 8}$ | $\mathbf{5 6}$ |
| 225 | 45 | 50 | 60 | 70 |

The only suitable case is highlighted in bold. As the total number of flowers increases, the numbers of individual types of flowers also increase, so higher unlisted options likely do not meet the above requirements. There are a total of 180 flowers in the bed in front of the school.

Note. Once we find a number in a row that is less than 30 or greater than 60, we do not need to calculate the remaining entries - this option certainly does not meet the requirements. Therefore, the previous discussion can be complete even though the table is incomplete.

Alternative solution. First, we calculate what fraction of all the flowers are primroses:

$$
1-\frac{1}{5}-\frac{2}{9}-\frac{4}{15}=\frac{14}{45}
$$

For the number of primroses to be a natural number, the total number of flowers must be a multiple of 45. In such a case, the numbers of other types of flowers are also natural numbers.

The numbers of all individual types are in the range from 30 to 60 (inclusive), from which we can determine the range of the total number of flowers:

Tulips make up $\frac{1}{5}$ of the total number of flowers, so the total number of flowers must be

$$
\text { from } 30 \cdot 5=150 \text { to } 60 \cdot 5=300
$$

According to the proportion of daffodils, the total number of flowers must be

$$
\text { from } 30 \cdot \frac{9}{2}=135 \text { to } 60 \cdot \frac{9}{2}=270
$$

According to the hyacinths, the total number of flowers must be

$$
\text { from } 30 \cdot \frac{15}{4}=112.5 \text { to } 60 \cdot \frac{15}{4}=225
$$

And according to the primroses

$$
\text { from } 30 \cdot \frac{45}{14} \approx 96.4 \text { to } 60 \cdot \frac{45}{14} \approx 192.9
$$

All four conditions must be met simultaneously, meaning the total number of flowers is in the range from 150 to 192 (inclusive). Among these numbers, there is only one multiple of 45, namely 180. There are a total of 180 flowers in the bed.