# 2008 iTest Problems/Problem 17

## Problem

One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy counts a total of $28$ heads belonging to the people, horses, and dog. She counts a total of $92$ legs belonging to everyone, and notes that nobody is missing any legs.

Upon returning home Wendy gives Alexis a little problem solving practice, "I saw $28$ heads and $92$ legs belonging to people, horses, and dogs. Assuming two legs per person and four for the other animals, how many people did I see?" Alexis scribbles out some algebra and answers correctly. What is her answer?

## Solution

Let $h$ be the number of humans, $p$ be the number of horses, and $d$ be the number of dogs. Write a system of equations (and one inequality). $\[h+p+d=28\]$ $\[2h+4p+4d=92\]$ $\[h Multiply the first equation by $4$ to get $\[4h+4p+4d=112\]$ Subtract the second equation to get $\[2h = 20\]$ $\[h = 10\]$ When the value is plugged in, we can find multiple integers $p$ and $d$ that satisfy the equations and inequality. Thus, there are $\boxed{10}$ people.