How do you pro-rate subscription upgrades?

Question

The situation:

A person has a subscription which allows them 3 licenses for 12 months @ $6500/12mo

After 3 months, they upgrade to a 5 licenses per 12-month plan @ $9000/12mo.

* They pay only the difference between the plans ($2500) to make a total of $9000 *

(the only 2 variables that change are the total/ price and the number of licenses)

How long does their plan now last?

Please SUPPLY THE EQUATION and NOT JUST THE ANSWER!

Bonus: Supply a formula to calculate what they'd pay to upgrade so their plan lasts EXACTLY 12 months.

Answer 1

They have used one quarter of the first plan, i.e. $6500/4 = $1625 worth, with $4875 remaining credit. This is allocated to the 5-licence plan together with an additional payment of $2500, so they are paying a total of $7375 on the new plan. This will pay for (7375/9000)×12 = 9 5/6 months, for a total time of 12 5/6 months.

Alternatively, you can say that they have now paid for the full year at the higher price, but for three months were only using the lower plan. So they should have 1/4 the difference available as further credit beyond the first year; that is, 1/4 (2500) = $625 extra credit. This will last for 625 / (9000/12) = 5/6 extra months, so the total period paid for is again 12 5/6 months.

For exactly 12 months, they should pay 1/4 ($6500) + 3/4 ($9000) = $8375 in total. Since they've already paid $6500, the additional payment should be $1875. Alternatively, it's easier to say that they should only pay 3/4 of the difference (since they'll only be using the higher plan for 3/4 of the year), so the additional payment should be 3/4 ($2500) = $1875.