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Use the Banker’s rule!

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? 

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Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
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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.