Starting with $2,000 on March 3, you deposit $500 23 days from March 3, withdraw $800 69 days from March 3, and deposit $600 121 days from March 3. If your final balance is $2,458 150 days from March 3, what is the simple interest rate for your account?

By the Banker's Rule, the simple interest amount is

I=Prt

with P the principal, r the annual interest, t the time in years, or fractions of a year. We are looking for r.

Let's first find the total interest for the 150 days, counting the $800 withdrawal as a negative deposit:

I=2458 - (2000+500-800+600) = 2458-2300 = 158

Let's compare this to the interest amounts obtained by the Banker's Rule:

I=2000*r*(150/365) + 500*r*(150-23)/365 - 800*r*(150-69)/365 + 600*r*(150-121)/365

We can factor out the unknown r out of each term and combine the rest to get

I=2044*r

Now set this equal to 158 and solve:

I=2044*r=158

r=0.0773, or 7.73% p.a.