Question

Peter is renting two kinds of rafts for a vacation company. One type seats 3

people and the other seats 5 people. If 53 people are using the vacation
company and he rents 13 rafts, how many of each type of raft does he rent?

Answer 1

Peter rents 6 of the rafts seating 3 people, and 7 of the rafts seating 5 people.

Explanation:

Let
`x` denote the number of rafts seating 3 people and
`y` the number of rafts seating 5 people. The total number of rafts is 13, so
`x+y=13`. We can also assume that every raft is filled to the brim (it isn't stated explicitly though, but it's probably the intention of the question maker), so
`3x+5y=53`.

It's always a good idea to put these equations under each other:

`\left \{ {{x+y = 13} \atop {3x+5y=53}} \right.`

We can subtract the first equation three times from the second one to obtain
`(3x+5y)-3(x+y) = 53 - 3(13)\\3x-3x + 5y - 3y = 53 - 39\\2y = 14\\y=(14)/(2) = 7`

Now, substitute this found value for
`y` into
`x+y = 13` and we see that
`x=6`. We are now done: Peter rents 6 of the rafts seating 3 people, and 7 of the rafts seating 5 people.