At the end of the day, February 14th, a florist had 120 roses left in his shop, all of which were red, white or pink in color and either long or short-stemmed. A third of the roses were short-stemmed, 20 of which were white and 15 of which were pink. The percentage of pink roses that were short-stemmed equaled the percentage of red roses that were short-stemmed. If none of the long-stemmed roses were white, what percentage of the long-stemmed roses were red?

We have been given that at the end of the day, February 14th, a florist had 120 roses left in his shop, all of which were red, white or pink in color and either long or short-stemmed.

We are also told that 1/3 of the roses were short-stemmed.

[tex]\text{Short-stemmed roses}=120\times \frac{1}{3}=40[/tex]

Since 20 of those were white and 15 of which were pink, so short stemmed red roses would be [tex]40-(20+15)=40-35=5[/tex].

Now, we will find number of long-stemmed roses by subtracting number of short-stemmed roses from total roses as:

[tex]\text{Long-stemmed roses}=120-40=80[/tex]

We are also told that none of the long-stemmed roses were white, so total number of white roses would be [tex]20+0=20[/tex].

Let p represent the number of total pink roses.

Now, total number of red roses would be total roses (120) minus total pink roses (p) minus total white roses (20).

[tex]\text{Total red roses}=120-p-20[/tex]

[tex]\text{Total red roses}=100-p[/tex]

We have been given that the percentage of pink roses that were short-stemmed equaled the percentage of red roses that were short-stemmed. We can represent this information in an equation as:

[tex]\frac{\text{Short-stemmed pink roses}}{\text{Total pink roses}}=\frac{\text{Short-stemmed red roses}}{\text{Total red roses}}[/tex]

[tex]\frac{15}{p}=\frac{5}{100-p}[/tex]

Let us solve for p by cross-multiplication:

[tex]1500-15p=5p[/tex]

[tex]1500-15p+15p=5p+15p[/tex]

[tex]1500=20p[/tex]

[tex]20p=1500[/tex]

[tex]\frac{20p}{20}=\frac{1500}{20}[/tex]

[tex]p=75[/tex]

Since total number of pink roses is 75, so total number of red roses would be [tex]100-75=25[/tex].

We already figured it out that 5 roses are short-stemmed, so long-stemmed roses would be [tex]25-5=20[/tex].

Now, we have long stemmed roses is equal to 20 and total long-stemmed roses is equal to 80.

Let us find 20 is what percent of 80.

[tex]\text{Percentage of the long-stemmed roses that were red}=\frac{20}{80}\times 100[/tex]

[tex]\text{Percentage of the long-stemmed roses that were red}=\frac{1}{4}\times 100[/tex]

[tex]\text{Percentage of the long-stemmed roses that were red}=25[/tex]

Therefore, 25% of the long-stemmed roses were red.