Track / Stage 5 / 94 of 400 #694 of 2000

Problem 694

AIME late

Combinatorics Difficulty 5.3 Find the answer

## 263. Math Puzzle $4 / 87$

In a parking lot, there are cars, mopeds, and motorcycles with sidecars. There are a total of 16 vehicles with 50 wheels (excluding spare wheels). There are as many cars as motorcycles with sidecars.

How many vehicles of each type are parked?

Your answer

Official solution

We denote passenger cars with $x$ , sidecar motorcycles with $y$ , and mopeds with $z$ and set up the following equations:

$\text { (1) } \quad x+y+z=16 ; \quad \text { (II) } \quad 4 x+3 y+21=50 ; \quad \text { (III) } \quad x=y$

Substituting (III) into (I) and (II) leads to $x=6$ .

There are 6 passenger cars. From (III), it follows that 6 sidecar motorcycles are parked on the lot. Therefore, 4 mopeds complete the "fleet".

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.