By the Banker's Rule, the simple interest amount is
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
Now set this equal to 158 and solve:
r=0.0773, or 7.73% p.a.