Problema Solution
Aunt Alice gave each of her three nieces a number of silver coins to equal 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 two other 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? (Hint: Start with each sister having 16 silver dollars and work backwards.)
Answer provided by our tutors
lets start with each sister having 16 silver dollars and work backwards
youngest = 16 ; middle = 16 ; oldest = 16
the oldest was to split half of her dollars equally between the two younger sisters meaning the oldest had 16*2 = 32 before this step while the youngest and the middle had 16 - (16/2) = 8
youngest = 8 ; middle = 8 ; oldest = 32
the middle sister would then give each of the others 4 silver dollars. if the middle sister takes back 4 dollars from each sister she will have 8 + 2*4 = 16, the youngest will have 8 - 4 = 4 and the oldest will have 32 - 4 = 28
youngest = 4 ; middle = 16 ; oldest = 28
the youngest would split half her silver dollars evenly with the two other sisters follows the youngest had double the amount that is
youngest = 2*4 = 8 ; middle = 16 - 2 = 14 ; oldest = 28 - 2 = 26
the sisters are 8, 14 and 26 years old.
Another ways to solve this problem is to start with the age of the sister.
let x, y and z be they age and x < y < z
start situation:
x, y ,z
the youngest would split half her silver dollars evenly with the two other sisters
x - x/2, y + x/4, z + x/4
the middle sister would then give each of the others 4 silver dollars
x - x/2 + 4, y + x/4 - 2*4, z + x/4 + 4
finally, the oldest was to split half of her dollars equally between the two younger sisters
x - x/2 + 4 + (1/4)(z + x/4 + 4) = 16
y + x/4 - 8 + (1/4)(z + x/4 + 4) = 16
(z + x/4 + 4)/2 = 16
by solving the system we find
x = 8 years
y = 14 years
z = 26 years
click here to see the step by step solution of the system of equations
the sisters are 8, 14 and 26 years old.