Huey, Dewey, Louie Numbers

Huey, Dewey, and Louie find that they must get from their home in Duckburg to a friend’s house in Possumtown, a distance of 52 km. Louie has a motorcycle, but it can manage only one passenger at a time in addition to the driver. This cycle can only go 20 kph at best. The three of them hit on a plan: Louie will take Huey part of the way, and then leave him to walk the rest of the way (at 5 kph). Then Louie will return to pick up Dewey, who has been walking all the while at 4 kph, and they will ride the rest of the way to Possumtown. They carried out their plan, which brought them all to their destination at exactly the same time. Now, can you figure out how many hours it took for the whole adventure?

Show/Hide Solution

x, y, and z are times.
x + y is the total travel time for each of H, D, and L.
z is the time during which L is going back to pick up D.
(Note: he travels more than 52 km.)
x + y is the number of hours it took for the whole adventure.

So we have 3 equations, ready to be solved by determinants:

H:	20x + 5y + 0z = 52	← (1)
D:	4x + 4z + 20y – 20z = 52
	→ 4x + 20y – 16z = 52	← (2)
L:	20x – 20z + 20y – 20z = 52
	→ 20x + 20y – 40z = 52	← (3)

Or, by hand:

(2)	4x + 20y – 16z = 52	→ x 5 →	20x + 100y – 80z	=	260
(3)	20x + 20y – 40z = 52	→ x -2 →	-40x – 40y + 80z	=	-104
			-20x + 60y	=	156
(1)			20x + 5y	=	52
			65y	=	208
			→ y	=	3.2

(1) 20x + 5(3.2) = 52 → 20x = 36 → x = 1.8

So x + y = 3.2 + 1.8 = 5 hours for the entire adventure.

We can also find z, not that we need it:

(2) 7.2 + 64 – 16z = 52 → 16z = 19.2 → z = 1.2