Answer:
11, 12.
Step-by-step explanation:
Let q represent number of quarters.
We have been given that Caroline has a maximum of 15 coins worth at least $2.85 combined. We are also told that Caroline has 3 dimes. This means that total coins are less than or equal to 15.
We can represent this information in an inequality as:
![q+3\leq 15...(1)](https://tex.z-dn.net/?f=q%2B3%5Cleq%2015...%281%29)
We are also told that the coins worth at least $2.85 combined. This means that the worth of all coins is greater than or equal to 2.85.
We know that each dime is worth $0.10 and each quarter is worth $0.25.
![0.25q+3(0.10)\geq 2.85...(2)](https://tex.z-dn.net/?f=0.25q%2B3%280.10%29%5Cgeq%202.85...%282%29)
Now, let us solve our system of inequalities.
From 1st inequality, we will get:
![q+3-3\leq 15-3](https://tex.z-dn.net/?f=q%2B3-3%5Cleq%2015-3)
![q\leq 12](https://tex.z-dn.net/?f=q%5Cleq%2012)
From 2nd inequality, we will get:
![0.25q+0.30\geq 2.85](https://tex.z-dn.net/?f=0.25q%2B0.30%5Cgeq%202.85)
![0.25q+0.30-0.30\geq 2.85-0.30](https://tex.z-dn.net/?f=0.25q%2B0.30-0.30%5Cgeq%202.85-0.30)
![0.25q\geq 2.55](https://tex.z-dn.net/?f=0.25q%5Cgeq%202.55)
![\frac{0.25q}{0.25}\geq \frac{2.55}{0.25}](https://tex.z-dn.net/?f=%5Cfrac%7B0.25q%7D%7B0.25%7D%5Cgeq%20%5Cfrac%7B2.55%7D%7B0.25%7D)
![q\geq 10.2](https://tex.z-dn.net/?f=q%5Cgeq%2010.2)
Upon combining our both inequalities, we will get:
![10.2\leq q\leq 12](https://tex.z-dn.net/?f=10.2%5Cleq%20q%5Cleq%2012)
This means that numbers of quarters would be greater than or equal to 10.2 and less than or equal to 12.
Since we cannot have 0.2 of a coin, therefore, Caroline could have 11 or 12 quarters.