This problem comparing two types of gym memberships — one with an up-front fee and a monthly fee and the other with just a monthly fee was motivated by a video at Khan Academy.
It should be useful for teachers and parents to follow up Khan videos with problems related to the video.
Question: A gym offers two type of memberships. The first membership has a $200 up-front fee and a $40 a monthly charge. The second membership has a $50 up-front fee and a $70 monthly charge. How many months must a person be a member of the gym in order for the first membership to be less expensive than the second membership?
Answer: M is the number of months the person is a member of the gym. The person is indifferent to the two memberships if
$200+$40M = $50+ $70M
Rearrange we get
$150 = $30 M
when M= 5 the total amount of money spent on the gym membership is $400 under both payment plans.
When M=4 the total expenditure on the first plan (with the $200 upfront fee) is $360. It exceeds cost of the second plan, $330.
For M=6 expenditures are $440 for the first plan and $470 for the second plan.