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Question: You need to rent a moving truck. You identified two companies that rent trucks. ABC-Rental charges $30 per day plus $0.15 per mile. U-Do-It Rental charges $15 per day plus $0.30 per mile. For how many miles will the daily cost of renting be the same?

Answer 1

Two important observations:

    1.  Although not explicitly stated, both rentals are for 1 day.

    2.  There are fixed costs (regardless of usage) and variable costs (cost per mile).

          The slope-intercept form of the equation of a line, y=mx+b, has b (the y-intercept) as the fixed costs and m (slope) as the variable costs.  Thus, we will write the equation in that form.

    3.  Considering units [which is extremely important,  (costs per mile)*(number of miles)=(variable costs)]

 

Let x = number of miles driven       [remember, everything is for 1 day]

 

So, the total costs are:

                    Total costs        =       Variable Costs                +          Fixed Costs

ABC-R:                C_A             =           $0.15x                        +              $30.00

U-Do-It:              C_U            =             $0.30x                       +              $15.00

 

"How many miles"  means to find x.

"the daily cost of renting be the same?"   means    C_A=C_U

              (once again, this is "daily renting cost")

 

Set C_A = C_U and solve for x:

        $0.15x + $30.00  =  $0.30x + $15.00

            -$0.15x +$30.00 = $15.00      [subtract $0.30 from both sides]

             -$0.15x = -$15.00             [subtract $30.00 from both sides]

              $0.15x = $15.00             [multiply both sides by (-1)]

                  x = 100 miles per day          [divide both sides by $0.15]

    [Note: Remember once again that this is per day amount]

  [Note: If you don't like decimal point values you could do:    15x+3000=30x+1500 ]

 

Checking (very important):

   Is  15x+3000=30x+1500   ?

        15(100)+3000=30(100)+1500   ?

                   4500 = 4500   ?yes