Answer:
The possible values for the number of quarters she could have is;
[tex]11\text{ and 12}[/tex]Explanation:
Given that Caroline has some dimes and some quarters.
she has a maximum of 15 coins worth at least $2.85 combined.
Let d and q represent the number of dimes and quarters she has.
1 dime = $0.10
1 quarter = $0.25
So, we have;
[tex]\begin{gathered} d+q\leq15\text{ ----------1} \\ 0.10d+0.25q\ge2.85\text{ ------2} \end{gathered}[/tex]if Caroline has 3 dimes;
[tex]d=3[/tex]substituting into equation 1 and 2;
[tex]\begin{gathered} d+q\leq15 \\ 3+q\leq15 \\ q\leq15-3 \\ q\leq12 \end{gathered}[/tex][tex]\begin{gathered} 0.10d+0.25q\ge2.85 \\ 0.10(3)+0.25q\ge2.85 \\ 0.30+0.25q\ge2.85 \\ 0.25q\ge2.85-0.30 \\ 0.25q\ge2.55 \\ q\ge\frac{2.55}{0.25} \\ q\ge10.2 \end{gathered}[/tex]Therefore;
[tex]10.2\leq q\leq12[/tex]So the possible values for the number of quarters she could have is;
[tex]11\text{ and 12}[/tex]