Applications of problems involving hourly rates, cell phone rates, and salary/commission are all similar in that they involve payment to someone for a service completed after a period of time has passed. For example, many employees are paid salaries based on the number of hours they have worked over a given period of time plus a commission, users of cell phones pay a phone bill based on the number of hours they talk on their phones during a given period of time, and many businesses charge hourly fees for you to use their products over a given period of time.
In order to solve these types of problems, it is necessary to
Suppose that the Aztec Cell Phone company charges $50 per month plus 15 cents per minute while the Southern Cell Phone Company charges no monthly fee but 25 cents per minute. After how many minutes of phone usage would a monthly phone bill be the same from both companies?
First, we need to write an equation
representing the cost/minute for each cell phone company. Let x
represent number of minutes.
for the Aztec Company will be:
The 50 represents the flat monthly charge and the 0.15x represents 15 cents for each minute (x) that someone uses the phone.
for the Southern Company will be:
There is no flat monthly charge, so the 0.25x represent 25 cents for each minute (x) that someone uses the phone.
To determine after how many minutes the two companies’ charge would be the same, we solve the two equations simultaneously.
Rewrite each equation
so that they are in the form
Multiply the bottom row by -1:
Add the two rows:
Divide both sides by -0.1:
To find the monthly cost, solve for y using either of the original equations:
Therefore, the cost for using each company’s cell phone will be equal after 500 minutes. The cost would be $125.
This information could also be obtained graphically by identifying the coordinates
of the intersection point
where the lines of our two equations cross each other.
The bold line
represents the graph
of the Aztec Company. The other line
represents the graph
of the Southern Company. The two lines intersect at the point
where both companies are charging the same amount of money for monthly phone use, $125. This occurs at 500 minutes.