How to find the number of kids who go to a theater when it is given the price of certain tickets?

Question

The problem is as follows:

Vincent goes to the movies with his kids, but when he wants to get tickets of $\$ 3.00$ each he observes in his pocket that he would lack money for two of their kids. As a result he ends up buying the $\$ > 1.5$ tickets in such a way that they all get in and he has $\$ 3.0$ left over. How many were his children?

The alternatives given in my book were:

$\begin{array}{ll} 1.&\textrm{8 kids}\\ 2.&\textrm{5 kids}\\ 3.&\textrm{7 kids}\\ 4.&\textrm{6 kids}\\ 5.&\textrm{9 kids}\\ \end{array}$

I'm some confused on how to approach this problem. What I thought to do was to equate the money he has at the beginning with what he has in the end.

I'm assuming that the money he has is used to purchase the tickets including him to see the movies along with this kids.

The later statement is not directly stated in the problem, but I think it is intended to imply. But is it okay to assume this?.

Therefore it would be:

Let $x$ the number of kids he has.

Therefore $3 + 3x - 3\cdot 2= m$

with $m$ to be the money he has in his pocket.

Then he he uses the tickets of 1.5 dollars each it would meant:

$1.5 + 1.5x + 3 = m$

By equating the both expressions it would meant:

$3 + 3x - 3\cdot 2 = 1.5 + 1.5x + 3$

Solving this yields:

$1.5x=7.5$

$x= 5$

Therefore it meant that the number of his kids is $5$, which it checks with what is mentioned in my answers sheet. But does it exist a different method to solve this?. I'm still dumbfounded if is it okay to assume that he purchases the tickets along with his kids or would he go with his kids and do not watch the movie?.

Answer 1

I would assume that Vincent has to buy a ticket, too, and that he pays the same as the kids. In my experience theaters charge less for kids than adults, but we only have prices that do not seem to depend on age. Then if $x$ is the number of kids he has to buy $x+1$ tickets. If he has $m$ in his pocket the first tells us $3(x+1)=m+6$ because he lacks the money for two kids at $3$ each. The second says $1.5(x+1)=m-3$ because he spends $3$ less than the money he has to get them all in. We have two equations in two unknowns. $$3(x+1)=m+6\\ 1.5(x+1)=m-3\\ 3(x+1)=2m-6\\ m+6=2m-6\\ m=12\\ x=5$$

This is equivalent to your approach. I agree the question is not clear about what Vincent spends to get himself into the movie. Your math is correct and you are reading the question the same way I am.