Rounding Errors

 
In the following 2 examples, the payoff date is August on a current year tax record (12% penalty, 7% interest, and 15% attorney fees).  Example 1 will show a penny more due after the payment is applied and example 2 will show a penny less due after the payment is applied.  These rounding errors can occur on accounts when a partial payment is applied to an account when penalty, interest, or attorney fees are involved.
 
Example 1:
 
base tax due  =                               $23,115.60
penalty due   = 23,115.60 x 0.12 = 2,773.872 = $ 2,773.87
interest due  = 23,115.60 x 0.07 = 1,618.092 = $ 1,618.09
                                              -----------
                                            = $27,507.56
attorney fees = 27,507.56 x 0.15 = 4,126.134 = $ 4,126.13
                                              -----------
total due (Payoff amount)                    = $31,633.69
 
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The taxpayer is making a payment amount of $2,634.69.
 
base tax paid = 2,634.69 / 31,633.69 x 23,115.60 = 1,925.23984 = $1,925.24
penalty paid  = 2,634.69 / 31,633.69 x  2,773.87 =   231.02861 = $  231.03
interest paid = 2,634.69 / 31,633.69 x  1,618.09 =   134.76662 = $  134.77
attorney paid = 2,634.69 / 31,633.69 x  4,126.13 =   343.65492 = $  343.65
                                                                ----------
total paid                                                     = $2,634.69
 
1,925.24 is applied to the base tax of 23,115.60 leaving 21,190.36.
 
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After the payment is applied, this will leave a base tax due of 21,190.36.
 
The new payoff amount for August:
 
base tax due  = 23,115.60 - 1,925.24 =          $21,190.36
penalty due   = 21,190.36 x 0.12 = 2,542.8432 = $ 2,542.84
interest due  = 21,190.36 x 0.07 = 1,483.3252 = $ 1,483.33
                                                -----------
                                              = $25,216.53
attorney fees = 25,216.53 x 0.15 = 3,782.4795 = $ 3,782.48
                                                -----------
total due                                     = $28,999.01
 
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If you take the original balance of 31,633.69 due and subtract out the payment of 2,634.69 you get 28,999.00.  As you can see from above the new balance is 28,999.01.  A difference of 0.01.  This is due to rounding.
 
Another way to look at this is to take the original balance of $31,633.69 due and subtract out the payment of $2,634.69 you get $28,999.00.  If you take the base tax of $21,190.36 and then add in the calculated penalty, interest, and attorney fees you get 28,999.01 due (a penny more).  So if you take the base tax due and subtract a penny you get $21,190.35 and then add in the calculated penalty, interest, and attorney fees you get 28,998.99 (1 cent short).
 
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Example 2:
 
base tax due  =                                $23,115.64
penalty due   = 23,115.63 x 0.12 = 2,773.8756 = $ 2,773.88
interest due  = 23,115.63 x 0.07 = 1,618.0941 = $ 1,618.09
                                                -----------
                                              = $27,507.61
attorney fees = 27,507.60 x 0.15 = 4,126.14   = $ 4,126.14
                                                -----------
total due (Payoff amount)                     = $31,633.75
 
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The taxpayer is making a payment amount of $2,634.62.
 
base tax paid = 2,634.62 / 31,633.75 x 23,115.64 = 1,925.18837 = $1,925.19
penalty paid  = 2,634.62 / 31,633.75 x  2,773.88 =   231.02287 = $  231.02
interest paid = 2,634.62 / 31,633.75 x  1,618.09 =   134.76278 = $  134.76
attorney paid = 2,634.62 / 31,633.75 x  4,126.14 =   343.64598 = $  343.65
                                                                ----------
total paid                                                     = $2,634.62
 
1,925.19 is applied to the base tax of 23,115.64 leaving 21,190.45.
 
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After the payment is applied, this will leave a base tax due of 21,190.45.
 
The new payoff amount for August:
 
base tax due  = 23,115.64 - 1,925.19 =          $21,190.45
penalty due   = 21,190.45 x 0.12 = 2,542.8540 = $ 2,542.85
interest due  = 21,190.45 x 0.07 = 1,483.3315 = $ 1,483.33
                                                -----------
                                              = $25,216.63
attorney fees = 25,216.63 x 0.15 = 3,782.4945 = $ 3,782.49
                                                -----------
total due                                     = $28,999.12
 
-----------------------------------------------------------
 
If you take the original balance of $31,633.75 due and subtract out the payment of $2,634.62 you get $28,999.13.  As you can see from above the new balance is $28,999.12.  A difference of $0.01-.  This is due to rounding.
 
Another way to look at this is to take the original balance of $31,633.75 due and subtract out the payment of $2,634.62 you get $28,999.13.  If you take the base tax of $21,190.45 and then add in the calculated penalty, interest, and attorney fees you get 28,999.12 (1 cent short).  So if you take the base tax due and add a penny you get $21,190.46 and then add in the calculated penalty, interest, and attorney fees you get $28,999.15 (2 cents more).
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