x = the number of minutes the phone is used
Plan A:
40¢ per minute ($0.40)
no other costs
Cost for 1 month = 0.4 x
Plan B:
$30 a month, even if you don't use the phone at all
100 free minutes
then 50¢ per minute ($0.50)
Cost for 1 month:
If 'x' is less than 100: Cost = 30
If 'x' is greater than 100:
Cost = 30 + 0.5(x - 100) (Because the first 100 minutes are free, and
you only pay for minutes past 100. There are [x-100] of those.)
Eliminate parentheses: Cost = 30 + 0.5x - 50
Combine like terms: Cost = 0.5x - 20
Which plan costs more ? It depends on how many minutes you use in a month.
If you use a small number of minutes, Plan A costs you more.
If you use a huge number of minutes, Plan B costs you more.
Where is the crossover point ? It's the number of minutes in one month
where the costs of both plans are equal.
If you use the phone for less than 100 minutes a month,
(where the cost of Plan B starts increasing with each minute):
0.4x = 30
Divide each side by 0.4: x = 75
Less than 75 minutes per month, Plan A costs less.
Past 75 minutes a month, Plan A costs more than $30, so Plan B costs less,
until Plan B starts charging for extra minutes.
If you use the phone for more than 100 minutes a month:
0.4 x = 0.5 x - 20
Add 20 to each side: 0.4 x + 20 = 0.5 x
Subtract 0.4x from each side: 20 = 0.1 x
Multiply each side by 10: 200 = x
There it is.
Now we can combine the results:
-- Less than 75 minutes in a month: Plan A costs less.
-- Between 75-200 minutes in a month: Plan B costs less.
-- More than 200 minutes a month: Plan A costs less again.
Complicated ? Absolutely ! That's why citizens' consumer groups are after
these companies, to try to get them to make their plans more understandable
to regular people. I know from personal experience that even a lot of the
salesmen in the phone stores could not figure this out and give you sound advice.