AUNT ALICE GAVE EACH OF HER 3 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 HER 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 2 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 THEN WORK BACK.
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
To solve this problem, we can use a step-by-step approach.
Let's start by assigning variables to represent the ages of the sisters. We'll use the variables A, B, and C for the oldest, middle, and youngest sister, respectively.
Step 1: After redistributing the money, each girl had 16 silver dollars.
This means that the total amount of money was distributed equally among the sisters. We can write this as an equation: A + B + C = 48 (16 * 3).
Step 2: The youngest sister split half of her money evenly with her sisters.
Since the youngest sister had 16 silver dollars after the redistribution, she must have originally had 16 + 16 = 32 silver dollars.
Step 3: The middle sister gave each of the others 4 silver dollars.
The middle sister had 16 silver dollars after the redistribution, but she gave each of the other sisters 4 silver dollars. This means that the youngest and oldest sisters received 4 silver dollars each from the middle sister. So, the youngest sister now has 32 + 4 = 36 silver dollars, and the oldest sister also has 16 + 4 = 20 silver dollars.
Step 4: The oldest sister split half of her dollars equally between the two younger sisters.
The oldest sister had 20 silver dollars after the redistribution, but she split half of her money equally with the two younger sisters. This means she gave 10 silver dollars to each of them. So, the youngest sister now has 36 + 10 = 46 silver dollars, and the middle sister has 16 + 10 = 26 silver dollars.
Step 5: Calculate the ages of the sisters.
From Step 1, we know that A + B + C = 48. Given the values we derived from the previous steps, we can substitute them back into the equation: 20 + 26 + 46 = 48.
Simplifying this equation, we find that A + B + C = 92.
Since the total age of the sisters is 92, we need to find three numbers whose sum is 92. By trial and error, we can determine that the ages of the sisters are: A = 36, B = 26, and C = 30.
Therefore, the youngest sister is 30 years old, the middle sister is 26 years old, and the oldest sister is 36 years old.
So, the answer is: The youngest sister is 30 years old, the middle sister is 26 years old, and the oldest sister is 36 years old.