College Algebra Help

Question

A social club charters a bus at a cost of $780 to take a group of members on an excursion to Atlantic City. At the last minute, five people in the group decide not to go. This raises the transportation cost per person by $1. How many people originally intended to take the trip?

Answer 1

Hi Alexis

 

let x =number of members originally signed up

 

let y = original price of ticket per member

 

So y = 780/x. (Total cost divided by number members)

 

but this changes when 5 members drop out

 

Now we are told that the price goes up 1 dollar 

 

so y+1 = 780/(x-5). (This is the new ticket price)

 

multiply both sides of this this equation by (x-5) and we get

 

(x-5)(y+1)=780

 

we know y=780/x from our first equation so we will replace y with this

 

(x-5)(780/x + 1)=780

 

now we multiply

 

780 + x - 3900/x -5 = 780. 

 

Now subtract 780 from both sides and we get the following 

 

x - 3900/x -5=0

 

Now multiply both sides of the equation by x to get rid of x in the denominator and we get

 

x^2  -5x -3900=0

 

now factor

 

(X+60)(X-65)=0

 

Therefore (X-65)=0 and X=65

 

There were originally 65 members who signed up for the trip. I 

 

 

Answer 2

Hello Alexis,

Let P = the price per person originally,

and N = the number of people originally.

 

Then NP = 780 {Eq 1}

→ P = 780 / N {Eq 2}

 

After 5 people drop out, 'N' goes down by 5 and P goes up by 1, but the cost of chartering the bus would remain the same,

→ (N - 5) * (P + 1) = 780,

→ NP + N - 5P - 5 = 780 (by FOILing)

→ 780 + N - 5(780 / N) - 5 = 780 (by using {Eq 1} and {Eq 2})

→ N - 5(780 / N) -5 = 0 (by subtracting 780 from both sides)

→ N² - 5(780) - 5N = 0 (by multiplying through the whole equation by 'N' to clear the fraction)

→ N² - 5N - 3900 = 0 (by putting in descending order of powers)

→ (N - 65)(N + 60) = 0 (by factoring)

→ N = 65 or N = -60

 

* N = -60 would not make sense for the story problem, 

so N = 65 is the original number of people intending to go.

 

I hope the above helps, and thank you for posting the question.

 

Fractions are a bit tough to follow when typing on this webpage.  If you would like a cleaner version, please feel free to message me directly, and I can send a PDF version.

 

Kind regards,

Michael E.