2440.21 – Duckburg to Possumtown


Huey, Dewey, and Louie must get from their home in Duckburg to a friend's house in Possumtown, a distance of 52 km. Louie has a motorcycle that can go 20 kph but no faster and can only manage a single passenger.

The three make a plan. Louie will take Huey part of the way and then leave him to waddle the rest (at 5 kph--very good speed for a duck). Then Louie will go back and pick up Dewey, who has been waddling along all this time (at 4 kph). The two of them will ride on to Possumtown. The plan works perfectly: all three arrive at their destination at the same time.

Tell me: how many hours did it take for them to get from Duckburg to Possumtown?


Solution

A chart for this problem is below. The three variables, x,y,zx, y, z, are times: xx is the time that Huey rides, yy is the time that Huey walks, and zz is the time that Louie takes to go back and pick up Dewey. Check that the distances in the chart are correct. Check that each duck travels for x+yx + y minutes so that this is the answer to the problem.

The algebraic sums of the distances each duck travels are:

H: 20x+5y=20x+5yD: 4(x+z)+20(yz)=4x+20y16zL: 20x20z+20(yz)=20x+20y40z.\begin{aligned}H&:& \text{ } 20 x + 5 y &= 20 x + 5 y \\D&:& \text{ } 4(x + z) + 20(y - z) &= 4x + 20 y -16z\\L&:& \text{ } 20x - 20z + 20(y - z) &= 20x + 20 y - 40 z.\end{aligned}

Each duck travels the same net distance of 52 km. This gives three equations in the three unknowns:

20x+5y+0z=524x+20y16z=5220x+20y40z=52.\begin{aligned}20 x + 5 y + 0 z &= 52 \\4x + 20 y -16z &= 52 \\20x + 20 y - 40 z &= 52.\end{aligned}

Solving this set of equations gives x=1.8,y=3.2,z=1.2x = 1.8, y = 3.2, z = 1.2. The answer is x+y=1.8+3.2=5x + y = 1.8 + 3.2 = 5 hours!