Sure, let's break down the problem step by step.
1. You are given that there are a total of 545 flowers in the garden.
2. Let's denote the total number of flowers by [tex]\( T \)[/tex] and the number of flowers that are roses by [tex]\( R \)[/tex].
Thus, [tex]\( T = 545 \)[/tex].
3. According to the problem, [tex]\(\frac{1}{S}\)[/tex] fraction of the flowers are roses. We need to find [tex]\( S \)[/tex].
Given that [tex]\( R = \frac{1}{S} \times T \)[/tex], substituting [tex]\( T \)[/tex] we get:
[tex]\[
R = \frac{1}{S} \times 545
\][/tex]
4. The rest of the flowers are not roses. The number of non-rose flowers, [tex]\( F \)[/tex], would be:
[tex]\[
F = T - R = 545 - \frac{545}{S}
\][/tex]
5. To determine [tex]\( S \)[/tex], we set up the following equation based on the given condition:
[tex]\[
S = \frac{545}{545 - 1}
\][/tex]
Simplifying the expression inside the fraction:
[tex]\[
S = \frac{545}{544} = 1.00183823529
\][/tex]
It indicates that there is approximately 1 over 544th of the flowers that are roses, suggesting that [tex]\( S \approx 544 \)[/tex].
6. Using this [tex]\( S \approx 544 \)[/tex], we find the number of the rest of the flowers.
[tex]\[
F = 545 - \frac{545}{544} \approx 545 - 1.00183823529 = 543.9981617647059
\][/tex]
So, the fraction number [tex]\( S \)[/tex] is approximately 544, and the number of the rest of the flowers is approximately 543.998.
Therefore, the step-by-step solution yields [tex]\( S \approx 544 \)[/tex] and the number of remaining flowers [tex]\( F \approx 543.998 \)[/tex].
