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

or M=5.

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.

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