Aunt alice gave each of her three nieces a number of silver dollars equal to their ages. the youngest thought it was unfair. they agreed to redistribute the money. the youngest would first split half of her silver dollars evenly with the other two sisters. then, the middle sister would give each of the others 4 silver dollars. finally the oldest split half of her dollars equally between the two younger sisters. after all of the exchanging was done each girl had 16 silver dollars. how old are the sisters?

Let the sisters have dollars in the amounts of a,b,c. 

After the youngest splits half of hers with the others, they have 

a/2, b+a/4, c+a/4 

After the middle gives each of the others 4, they have 

a/2 + 4, b+a/4 - 8, c+a/4 + 4 

After the eldest splits half of hers with the others, they have 

a/2 + 4 + (c+a/4 + 4)/4, b+a/4 - 8 + (c+a/4 + 4)/4, (c+a/4 + 4)/2 

Now they all have 16. 

a/2 + 4 + (c+a/4 + 4)/4 = 16 
b+a/4 - 8 + (c+a/4 + 4)/4 = 16 
(c+a/4 + 4)/2 = 16 

substituting in the last expression, we have 

a/2 + 4 + 8 = 16 
b + a/4 - 8 + 8 = 16 
c/2 + a/8 + 2 = 16 

a = 8 
b = 14 
c = 26