Word Problems: Rates, Salaries, and Commissions

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

know how to solve systems of linear equations
know how to work with systems of equations using the graphing calculator
know how to work with a table of values for a function using the graphing calculator
know how to graph linear equations and find their intersections
know how to write and solve an equation for a word problem

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.

The equation 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.

The equation 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:

minutes

To find the monthly cost, solve for y using either of the original equations:

dollars

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.

Aztec Company:

Southern Company:

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.

Example Group #1

No audio files were recorded for this set of examples.

	The Aztec Cell Phone company charges $50 per month plus 15 cents per minute. The Southern Cell Phone Company charges no monthly fee but 25 cents per minute. Susan is trying to determine which company’s phone is right for her. She cannot afford to pay more than $60 per month for her phone. Which should she choose?

	What is your answer?

	Tara is a sales representative for a cosmetic company. She is paid $5.15 per hour each week plus a commission of 10% of the amount of sales over $5000. She works 40 hours one week, and she sells $7260 worth of cosmetics during that week. She has been offered a job for another cosmetic company that pays $5.00 per hour for a 40-hour work week plus a commission of 4% of total sales. Which job would pay more? Should she change jobs?

	What is your answer?

Example Group #2

No audio files were recorded for this set of examples.

	Jake’s Surf Shop rents surfboards for $6.00 plus $3.00 per hour. Rita’s rents them for $9.00 plus $2.50 per hour. Which is cheaper for which time intervals? Rentals less than 6 hours are cheaper at Rita’s; Rentals more than 6 hours are cheaper at Jake’s. Rentals are always cheaper at Jake’s. Rentals less than 6 hours are cheaper at Jake’s; Rentals more than 6 hours are cheaper at Rita’s. Rentals are always cheaper at Rita’s.

	What is your answer?

	Two furniture salesmen are comparing their salaries. Tom is paid $5.00 per hour plus a 15% commission of his total sales. Bob is paid $9.00 per hour plus a 10% commission of his total sales. Suppose each has sold $5000 worth of furniture, who would make the most money and during which time intervals? Tom always earns more than Bob. Bob always earns more than Tom. Bob earns more up to 62.5 hours; Tom earns more after 62.5 hours Tom earns more up to 62.5 hours; Bob earns more after 62.5 hours

	What is your answer?