Suppose you deposit $500 into an account with 7.8% interest compounded annually. how long will it take for the account to be worth $1000

7.8% from 500 is:

500 * 7.8% =
= 500 * 78/1000 =
= 5 * 78/10 =
= 1 * 78/2 =
= 78/2 =
= 39

To get $1000 from $500, we need extra $500, so we have to find out how many 39s can fit in those $500:

500 / 39 = 12 and the remainder of 32

So generally you will need about 12 + 1 = 13 years to make the account worth $1000.

1/x-1+3/x-1=1
x is -1.333