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 redistributed the money. the youngest would split half her silver dollars evenly with the two sisters. the middle sister 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?
start with each sister having 16 silver dollars and work backwards
To solve this problem, we'll start with each sister having 16 silver dollars and work backwards.
Let's assign variables to represent the ages of the three sisters:
- Let's say the age of the youngest sister is Y
- The age of the middle sister is M
- The age of the oldest sister is O
1) "The youngest would split half her silver dollars evenly with the two sisters."
Since the youngest sister had 16 silver dollars after the redistribution, she had received 4 (which is half of 8) silver dollars from each of her sisters. Hence, her original number of silver dollars must have been 8. So, Y = 8.
2) "The middle sister would then give each of the others 4 silver dollars."
Since the middle sister had 16 silver dollars after the redistribution, she gave each of the others 4 silver dollars. So, originally, she had 8 + 4 + 4 = 16 silver dollars. Therefore, M = 8 + 4 + 4 = 16.
3) "Finally, the oldest was to split half of her dollars equally between the two younger sisters."
Since the oldest sister had 16 silver dollars after the redistribution, she split half of her dollars (8 silver dollars) equally between the two younger sisters. Hence, each younger sister received 4 silver dollars from the oldest sister.
Considering the original distribution, the youngest sister had 8 silver dollars, and she received 4 silver dollars from the middle sister and 4 silver dollars from the oldest sister. So,
Y = 8 + 4 + 4 = 16
Y = 16
Therefore, the youngest sister is 16 years old.
Knowing that M = 16, and the middle sister gave 4 silver dollars each to both the youngest and the oldest sister, her original number of silver dollars would have been:
M = 16 + 4 + 4 = 24
Therefore, the middle sister is 24 years old.
Finally, we need to find the age of the oldest sister (O).
We know that the oldest sister had 16 silver dollars after the redistribution. Also, she split half of her dollars (8 silver dollars) equally between the two younger sisters. So, each younger sister received 4 silver dollars from the oldest sister. Therefore,
O = 16 + 4 + 4 = 24
Therefore, the oldest sister is 24 years old.
In conclusion, the ages of the sisters are:
- The youngest sister is 16 years old.
- The middle sister is 24 years old.
- The oldest sister is also 24 years old.