Find the equation!

Question

A number of students charter a bus to go to a school football game at a total cost of $80. Eight of the students are ill and cannot go, so each remaining student ends up paying an extra 50cents. Set up an equation that will help to find the number of students that went on the bus.

Answer 1

First take inventory of the things we know:

 

cost = 80.00   <--- We do not need the decimals, but we are dealing with money so we keep this format.

students = x   <--- This information was not given, so we use a variable in its place.

cost per student = y   <--- This is the cost that each student would have been required to pay; not given in the problem.

sick students = 8

extra cost = 0.50   <--- The 50 cents that the remaining students will have to pay in addition to the cost per student.

 

The total cost of the trip is being paid by charging each student an equal amount of money. We can create a skeleton equation by using the units, then insert the chosen variables from above.

 

• students * (cost/student) = total cost   <--- Keep in mind that (cost/student) is read as "price per student".
• x*y = 80   <--- Insert the chosen variables and constants above.

• (students - sick students) * (cost/student + extra cost) = total cost   <--- Add in the extra information.
• (x - 8)*(y+0.50) = 80   <--- The last step! Just replace the words with the variables and constants(numbers).

 

 From the first equation

• y = 80/x

We use this information and substitute into equation two

 

• (x - 8)(80/x+0.5) = 80
• 80 + 0.5x - (640/x) - 4 = 80   <--- Distribute
• 0.5x - (640/x) = 4   <--- Combine like terms
• 0.5x^2 - 640 = 4x   <--- Multiply both sides by x
• 0.5x^2 - 4x - 640 = 0   <--- Subtract 4x from both sides

 

Now that we have a quadratic equation in standard form (ax^2 + bx + c = 0), we can use the quadratic equation.

 

We should get x = -32 or x = 40. We cannot have negative students, so we know that the x = 40 is the correct answers.

 

By using the first equation we find:

 

• x*y = 80.00
• y = 80.00/x
• y = 80.00/40
• y = $2.00

 

This is the cost each student had to pay prior to the other students getting sick. If we add the extra 50 cents we get the new cost.

 

We found:

 

1. There were 40 total students, but only 32 went on the trip because 8 were sick.
2. Each student originally had to pay $2.00, but the cost was raised to $2.50.

 

Answer 2

We are given that a number of students are going to a football game, but don't know how many. Let's call this number of students, n. Let's also assume that the students split the total cost evenly, so that each student pays a price, p. 

 

Then we have,

 

1. (n)(p) = $80

 

However, 8 of our students are unable to go, leaving us with (n-8) students. As a consequence, the price each student must pay has increased by $0.50, such that each remaining student must now pay (p + 0.50).

 

Then we also have, 

 

2. (n-8)(p+0.50) = $80

 

We now have a system of 2 equations with 2 variables, n and p, which indicates that we can use this information to solve for both. Since we are solving for the number of students, we should substitute p with an expression for n to have one equation and one unknown. 

 

Rearranging equation 1, we have

p = 80/n

 

Substituting this into equation 2,

(n-8)((80/n) +0.50) =80

 

Distributing and simplifying, 

80 - 640/n + 0.50n - 4 = 80

-640/n + 0.50n-4 = 0

 

We notice that our equation has terms with coefficients in front of n to the power -1, 1 and 0 respectively. This hints that we can multiply the entire equation by n to formulate a quadratic equation to solve for n.

 

Thus, 

 

0.50n² -4n - 640 = 0.

 

Using the quadratic formula, we can solve the equation and obtain

n = 40 and n = -32.

 

n =-32 is our illogical answer, since the number of students cannot be negative and n=40 is our logical answer. 

 

BUT WE ARE NOT DONE! 

 

We need to make sure that our variable, n, matches what the problem is asking for. n represented the total number of students BEFORE the students dropped out. The question wants us to solve the number of students that went on the bus, which was (n-8). 

 

Therefore, the number of students that went on the bus was (40-8) = 32 students.

 

(Each remaining student now pays $2.50 instead of the $2.00 each was to pay when all 40 students were to attend.) 

 

 

 

 

 

 

Answer 3

Let x = the original number of students who planned to go and let y = the number of students who actually go. We can write a system of equations to model this situation:

 

y = x - 8        (there were 8 students who couldn't go)

80/x + 0.5 = 80/y        (each student would have paid 80/x, but they must now pay $0.50 more each)

 

 

 

Those are your two equations. If you actually want an answer, we'll first simplify that second equation a bit:

 

80/x + 0.5x/x = 80/y

(80 + 0.5x)/x = 80/y

(80 + 0.5x)y = 80x

 

Now, plug the first equation into the second and solve for x:

 

(80 + 0.5x)(x - 8) = 80x

80x - 640 + 0.5x² - 4x = 80x

-640 + 0.5x² - 4x = 0

-1280 + x² - 8x = 0

x² - 8x - 1280 = 0

(x - 40)(x + 32) = 0

x - 40 = 0    OR    x + 32 = 0

x = 40    OR    x = -32

 

Since we can't have a negative number of students, there were x = 40 students initially planning to go. After 8 got sick, there were 40 - 8 = 32 students who actually went.

 

 

 

Let's check our work.

 

Initially, 40 students would have paid 80/40 = $2.00 per person. 32 students will have to pay 80/32 = $2.50, which is $0.50 more per person. That checks out!