posted by mike on .
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