Question

A house was haunted by a combined total of 51 ghosts, goblins and ghouls. On Friday, there were half as many ghosts as there were goblins. On Saturday, two-thirds of the ghouls each became a ghost. On Sunday, 11 of the ghosts each became a goblin, and the ratio of ghouls to goblins became 1:3. If no other changes occurred, how many ghosts are there?

Answer 1

8 ghosts on Friday; 26 ghosts on Saturday; 15 ghosts on Sunday

Explanation:

Let x, y, z represent the numbers of ghosts, ghouls, and goblins on Friday. Their total is 51, and we know x = z/2.

On Saturday the number of ghosts is x +2/3y. and the number of ghouls is y -2/3y = 1/3y. The number of goblins remains unchanged at z.

On Sunday, the number of ghosts is x+2/3y-11; the number of ghouls is still 1/3y; and the number of goblins is z+11. The ratio of ghouls to goblins is 1:3, so ...

... (1/3y) : (z+11) = 1:3

... y = z+11

From this last relationship, and from those that existed on Friday, we can write 3 equations in the 3 unknowns.

• x +y +z = 51
• 2x -z = 0
• y -z = 11

Adding the first equation to each of the last two gives ...

... 3x +y = 51

... x +2y = 62

Subtracting the second from twice the first of these gives ...

... 2(3x +y) -(x+2y) = 2(51) -(62)

... 5x = 40

... x = 8

... 51 -3x = y = 27

... y -11 = z = 16

The number of ghosts changes daily, starting at 8 on Friday, going to 26 on Saturday, and ending at 15 on Sunday.