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The commission would be $4,550 and the high limit of income before deducting taxes and commission $24,550.

The solution of the alternate case, where the commission is payable on the net earnings after applying the normal income tax, but before deducting the excess-profits tax, is equally simple, viz:

Income before deducting commission or federal taxes..
Preliminary amount of income tax...

$70,000.00

4,620.00

Amount on which preliminary commission is to be computed. ... $65,380.00 Preliminary amount of commission @ 20%.....

13,076.00

Inasmuch as for every $100 of commission actually payable the income tax proper will be decreased by $6.00 and for every $6.00 decrease in the tax the commission will be $1.20 higher, it follows that the above preliminary amount of commission ($13,076) rep

$13,076 resents 98.8% of the correct amount, which is therefore

98.8 $13,234.82. This will leave a taxable income of $56,765.18 on which the tax will be:

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Commission will therefore be payable on $70,000 less $3,825.91 = $66,174.09 and at the rate of 20% will amount to $13,234.82, as above.

ct

Also here the formula X = 100

can be applied, (t)

100

being in this instance 10% of the 60% remaining after the 40% excess-profits tax has been deducted from the taxable income, or

20 X 6 X = 100

1.2 = 98.8

= 100

100

100-C

100

Likewise the limit beyond which the taxable income will become subject to the tax under the second bracket can be found with the formula previously used (C-T divided by -). The amount

5 on which the commission would have to be paid would accordingly $20,000

$1,620 be

= $22,975. The commission would then be .80 $4,595, and the high limit of income before deducting taxes and commission $24,595.

PROBLEM III COMMISSION PAYABLE AFTER DEDUCTING A PORTION OF THE FEDERAL

TAX It may also happen that commission is to be computed after deduction of a portion of the excess-profits or income tax or both. The procedure is in principle the same as that followed in the previous examples. For instance, in case commission is to be computed on the net profit after deducting two-thirds of the entire amount of the federal tax the method will be as follows: Income before deducting taxes and commission....

$70,000.00 Deduct two-thirds of preliminary amount of taxes, viz:

X $26,420 =

17,613.33 3 Amount on which preliminary commission is to be computed.... $52,386.67 Preliminary commission at 20%.

10,477.33 Inasmuch as for every $100 commission actually payable the tax will be decreased by $46 and for every $30.666 (or two-thirds of $46) of decrease in the tax, the preliminary commission will be increased by $9.20, it follows that for every $100 commission actually payable the preliminary commission will be increased by x $9.20, or $6.133. Consequently, the preliminary commission ($10,477-33) is 93.863% of the correct amount and the latter amounts to $10,477.33

= $11,161.93. This will leave a taxable income of .9386666

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Total federal tax

$21,285.51 Commission will therefore be payable on $70,000, less two thirds of $21,285.51 = $14,190.34, or on $55,809.66, and will, at the rate of 20%, amount to $11,161.93 as above. The above percentage of 93.863 can be obtained by the formula

ct

in which t is two thirds of the tax rate (46%)

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100

or 303%.

In the same manner will the formula 1/5 C - f T divided by

100-C

100

lead to a quick determination as to whether the tax rate under the second bracket should be considered or not. The highest amount, leaving the tax within the first bracket, on which the commission would have to be paid would accordingly be

2

$20,000 X $3,420 $17,720
3

= $22,150
.80

.80 The commission thereon would be $4,430 and the high limit of income before deducting taxes and commission would be $24.430.

After the foregoing demonstrations there should be no difficulty in solving the majority of the problems arising when commissions are to be computed after deducting federal taxes in part or in total. It should now also be plain that their solution can be attained by comparatively simple arithmetical methods which lie within the grasp of average intelligence, and that no higher or lower algebra nor infinite series need enter into it.

PROBLEM IV COMMISSION PAYABLE ON A SLIDING SCALE AFTER DEDUCTING THE ENTIRE AMOUNT OF BOTH EXCESS-PROFITS AND INCOME TAXES

Sometimes the rate of commission payable is on a sliding scale, for instance: 20% on the first...

$5,000 15% on the next.

5,000 10% on the next..

5,000 772% on the next.

3,000 5% on the next.

3,000 2% on the balance.

Also these cases can be solved by the general procedure used in the solution of the preceding problems, viz.:

(a) Preliminary computation of commission Income before deducting taxes and commission.

$70,000.00 Commission: 20% on $ 5,000.

$1,000 15% on $ 5,000.

750 10% on $ 5,000.

500 772% on $ 3,000.

225 5% on $ 3,000.

150 2% on $49,000...

980

$70,000

$3,605.00

Balance

$66,395.00 (b) Computation of the tax on the basis of $66,395 taxable income

Tar Excess-profits credit

$11,000.00 Portion of income taxable at 20%...

9,000.00 $ 1,800.00 Portion of income taxable at 40%...

46,395.00 18,558.00

Income

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(c) Calculation of correct amount of taxes and commission

Inasmuch as for every $100 taxes actually payable, the commission will evidently be $2.00 less, which in turn will decrease the tax by $.92, it follows that the above preliminary amount of taxes is

$24,761.70 99.08% of the correct amount, which is therefore

.9908 = $24,991.62. This will leave an amount of $45,008.38. on which commission is payable as follows:

On the first $21,000.00 on the basis of the sliding scale.....
On the balance of $24,008.38 @ 2%....

$2,625.00

480.17

$3,105.17

After deducting this commission from the $70,000 earned, the amount of $66,894.83 will represent the taxable income, on which the tax will be:

Income

Tax Excess-profits credit .....

$11,000.00 Portion of income taxable at 20%...

9,000.00 $ 1,800.00 Portion of income taxable at 40%...

46,894.83 18,757.93

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obtain the above percentage of 99.08 if (c) signifies the rate of commission payable on the residuary amount of the profits, viz., 2%.

In case commission is payable on a sliding scale the value of (c) is not always self-evident, nor is the tax rate (t) always readily determined. The problem is to find the limits beyond which the earnings before deducting taxes and commission would become

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