In the US, at least, "rent" is a monthly fee you pay for the use of something (whether it be an apartment, a car or a carpet shampooer). When you're done with the item, it is returned to the owner. When you "lease" something, the monthly payments you make are applied to the total cost, and you have the option of eventually purchasing the item. For example, if you lease an automobile, you might pay a monthly fee for the two-year term of the lease. When the lease expires, you have the option of returning the car, or buying it. If you decide to buy it, the payments you have made thus far will be deducted from the total cost.
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