Question

A group of psychologists organized a research study on a college campus. Participants were invited for the study and they were paid $15$ if they were students and $10$ if they were non-students. If there were 10 participants and the college pays a total cost of $120$, how many of the participants were students.

Answer 1

Hint: We can assume the number of student participant and non-student participants as 2 variables. Then we can form an equation with the total participants. Another equation can be formed using the money the college pays. Then we can solve the equations to get the required number of student participants.

Complete step by step answer:

Let x be the number of student participants and y be the number of non-student participants.
It is given that the total number of participants is 10.
$ \Rightarrow x + y = 10$.. (1)
It is given that $15$ is paid for student participants and $10$ for non-student participants. Then its sum is given by,
$15x + 10y$
It is also given that a total of $120$ is paid by the college.
$ \Rightarrow 15x + 10y = 120$
On dividing throughout with 5, we get,
$ \Rightarrow 3x + 2y = 24$… (2)
Now we can solve the equations (1) and (2). For that, we multiply (1) with 2 and subtract it from (2)
\[
   \Rightarrow 3x + 2y = 24 \\
  \underline {\left( - \right)2x + 2y = 20} \\
   \Rightarrow x + 0 = 4 \\
 \]
$ \Rightarrow x = 4$
Therefore, the number of student participant is 4.

Note: Alternate method to solve the equations is by substitution.
We have equation $x + y = 10$ . On rearranging, we get
$y = 10 - x$.. (a)
We have the other equation as $3x + 2y = 24$
We can substitute equation (a), we get
$ \Rightarrow 3x + 2\left( {10 - x} \right) = 24$
On expanding the bracket, we get
$ \Rightarrow 3x + 20 - 2x = 24$
On further simplification, we get
$ \Rightarrow x = 4$
Therefore, the number of student participant is 4.