Aunt Alice gave each of her three nieces a number of silver dollars equal to their ages.The youngest felt that this was unfair. They agreed to redistribute the money. The youngest would split half her silver dollars evenly with the other sisters. The middle sister would would then give each of the others 4 silver dollars. Finally the oldest was to split half of her dollars equally between the two younger sisters. After exchanging the money each girl had 16 silver dollars. How old were the sisters? (Hint: Start with each sister having 16 silver dollars and work backwards.)

To solve this problem, let's start by working backwards. We're told that after exchanging the money, each girl had 16 silver dollars. So, let's start with each sister having 16 silver dollars and work through the exchange process.

Let's denote the ages of the sisters as A, B, and C, where A is the oldest, B is the middle sister, and C is the youngest.

According to the problem, the youngest sister split half her silver dollars evenly with the other sisters. So, if she had 16 silver dollars, she must have had 32 silver dollars before splitting. This means she is 32 years old (since Aunt Alice gave each niece a number of silver dollars equal to their ages).

Now, we know that the youngest sister had 32 silver dollars before splitting, and she split half of it evenly with the other sisters. This means that both the middle sister and the oldest sister received 16 silver dollars each from the youngest sister.

Next, the middle sister gave each of the others 4 silver dollars. Since she had 16 silver dollars, she must have initially received 8 silver dollars from both the youngest and oldest sisters (because she gave 4 silver dollars to both of them). This means that before the exchange, she had 8 + 16 = 24 silver dollars, which implies she is 24 years old.

Finally, the oldest sister split half of her dollars equally between the two younger sisters. So, after giving away half of her silver dollars, she had 16 silver dollars left. This means that before the exchange, she had 32 silver dollars (since she split it equally between the two younger sisters and they each received 16 silver dollars). Therefore, the oldest sister is 32 years old.

To summarize:
- The youngest sister is 32 years old.
- The middle sister is 24 years old.
- The oldest sister is 32 years old.

So, the ages of the three sisters are 32, 24, and 32, respectively.