Stuck on an algebra word problem, and can't seem to shake it. Could someone help me?

Question

Jeremy is looking at two different options for a new gym membership.

 

Plan A has a fee of $40 plus an additional fee of $19.95 per month. With this plan, the first 3 months are free.

 

Plan B has a fee of $59.85 plus an additional fee of $29.95 per month. With this plan, the first 6 months are free.

 

After how many months would both options cost the same amount?

Answer 1

Oren, just today I told a friend that the hardest part of a word (story) problem is not the math; it is the translation from the English phrases into the precise, concise math expressions (variables and operations).  So, you’re not alone.

First, let’s assign variables:
    A = the cost of Plan A
    B = the cost of Plan B
    M = the month that we’re interested in
 
Now, translate:
    “a fee of $40 plus an additional fee of $19.95 per month.” 
                Means      $40 +$19.95*M
    “the first 3 months are free”    means start at month 4
    “a fee of $59.85 plus an additional fee of $29.95 per month”
                Means      $59.85 + $29.95*M
      “the first 6 months are free”     means start at month 7
     “After how many months would both options cost the same amount?”
               Means set   A = B  so that you can report M

Hard part done, Now, the math:
         A  = $40 + $19.95*( |M-3| )       [ absolute value; first 3 months are free ]
         B  = $59.85 + $29.95*( |M-6| )    [ absolute value; first 6 months are free ]
          A = B

          $40 + $19.95*( |M-3| )   =  $59.85 + $29.95*( |M-6| )    
          40 + 19.95M – 59.85 = 59.85 + 29.95M – 179.70      (M>4 or M>6 )
          100 = 10M
          M = 10

     This means that at the 10th month Plan A costs the same as Plan B  (note: This problem uses what I call “magic numbers” that are inserted into the problem to make the calculations come out nicely – who ever heart of a base rate of $59.85 ??)

To check (very important):
  Plug in 10  to get
       M=10     A=$179.65    B=$179.65

Special note: the absolute values of this problem present an interesting graph.  Plan A starts off less expensive then becomes more expensive between months 4 up to 10, then becomes less expensive again.  I calculated these rates and plotted the curve (then translated it to characters): 

Month   Plan A    Plan B
1          $40.00           $59.85
2          $40.00           $59.85
3          $40.00           $59.85
4          $59.95           $59.85
5          $79.90           $59.85
6          $99.85           $59.85
7        $119.80          $89.80
8        $139.75        $119.75
9        $159.70        $149.70
10      $179.65        $179.65
11      $199.60        $209.60
12      $219.55        $239.55
13      $239.50        $269.50
14      $259.45        $299.45
15      $279.40        $329.40

=============================B=======A=========
===========================B====A==============
=========================B=A===================
=====================A=B=======================
================A====B=========================
BBBBBBBBBABBBBBBBB================================
AAAAAA==========================================
===============================================

Answer 2

Hi Oren,

     If we look at this over the course of a year here is what happens:

(Rounded up)

Month 1:

Plan A: $0

Plan B: $0

 

Month 2:

Plan A: $0
Plan B: $0

 

Month 3:

Plan A: $0
Plan B: $0

 

Month 4:

Plan A: $60

Plan B: $0

 

Month 5:

Plan A: $80

Plan B: $0

 

Month 6:

Plan A: $100

Plan B: $0

 

Month 7:

Plan A: $120

Plan B: $90

 

Month 8:

Plan A: $140

Plan B: $120

 

Month 9:

Plan A: $160

Plan B: $150

 

Month 10:

Plan A: $180

Plan B: $180

 

We can then say by month 6, plan A is already ahead by $159.70 while plan B is at 0.  

Therefore:

 

19.95X + 159.70 = 29.95X + 59.85

 

10X = 99.85

 

X = 9.99 

X = 10

 

Answer: Month 10

 

Hope this helps!