Help set up a word problem

Question

A $6 toll is charged to cross the bridge from the mainland to Sanibel Island. A six month pass, costing $50 reduces the toll to $2. A one year pass, costing $400 allows for free crossings. How many crossings per year does it take, on average, for the two six month passes to be the most economical chaice? Assume a constant number of trips per month.

Answer 1

Okay, you have THREE possible ways to pay, so first let's describe those with equations.

 

1. Pay as you go...12 months, times $6 per trip, would be 72X, where X is the number of trips per month
2. Buy the two $50 passes....$100 (two passes a year) + 2(12 months)X
3. Flat $400

 

Just the equations:

 

1. 72X
2. 100+24X
3. 400

 

We need to set up two inequalities, comparing that desired payment plan (two 6 month passes) to the others

 

First, compare to the $6 option

 

72X<100+24X (this will tell us when the pay as you go rate is a better bargain)

48X<100

X<2.08

 

Therefore, with 1 or 2 trips per month, you are better off just paying the $6.

 

Second, compare the two passes to the flat $400

 

100+24X<400

24X<300

X<12.5

 

Therefore once you pass 12 trips per month, you are better off paying the $400.

 

The two 6 month pass deal is best if your monthly trips are more than 2, but less than 13.

 

2<X<13

 

I used integers rather than the 2.08 and the 12.5, since we are talking about trips across a bridge.