# Paulie Gone has a shiny red wagon. When he puts his puppy, Hypotenuse, in the wagon, the wagon weighs in at 40 pounds. When Paulie rides the wagon by himself, the wagon weighs in at 64 pounds. Paulie weighs 3 times as much as Hypotenuse, how much does the empty wagon weigh?

Question
Upper Level Math
Paulie Gone has a shiny red wagon. When he puts his puppy, Hypotenuse, in the wagon, the wagon weighs in at 40 pounds. When Paulie rides the wagon by himself, the wagon weighs in at 64 pounds. Paulie weighs 3 times as much as Hypotenuse, how much does the empty wagon weigh?

2021-02-26
Let x be Hypotenuse’s weight so that Paulie weighs 3x. Let y be the weight of the empty wagon.
The puppy on the wagon weighs 40 pounds so:
$$\displaystyle{x}+{y}={40}{\left({1}\right)}$$
Pauli and the puppy on the wagon weighs 64 pounds so:
$$\displaystyle{3}{x}+{x}+{y}={64}$$
$$\displaystyle{4}{x}+{y}={64}{\left({2}\right)}$$
Solve for yy, the weight of the empty wagon. Using (1), solve for xx in terms of yy to obtain (3):
$$\displaystyle{x}={40}−{y}{\left({3}\right)}$$
Substitute (3) to (2) and solve for yy:
$$\displaystyle{4}{\left({40}−{y}\right)}+{y}={64}$$
$$\displaystyle{160}−{4}{y}+{y}={64}$$
$$\displaystyle{160}−{3}{y}={64}$$
$$\displaystyle−{3}{y}=−{96}$$
$$\displaystyle{y}={32}$$
So, the empty wagon weighs 32 pounds.

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