Bonus computations;

              2 HASKINS & SELLS April
Bonus Computations
WHEN certain companies made arrange-ments
to share profits, in the form of a
bonus, with employes, and designated for
the purpose of arriving at the amount a per­centage
to be applied to the net income, they
did not have sufficient regard for the possi­bility
of income and excess profits taxes.
With the income tax as one unknown
quantity and the bonus as a second unknown
quantity, both when ascertained to be consid­ered
as expenses, two formulae, as presented
below, have been worked out by our tax de­partment.
A reliable formula, involving a
third unknown quantity, namely, the excess
profits tax, has not yet been worked out.
Formula for computation of bonus based
on net income remaining after deduction of
income tax and before bonus is deducted.
I = Net income before deducting Bonuses
and Income Taxes.
B = Bonuses.
T = Income Taxes.
B = 5%(I— T)
T = 6% (I — B)
B = 5% [I — (6% [I — B]) ]
B = 5% [I — 6% I + 6% B]
B = 5% I —5% X 6% I + 5% X 6% B
B = 5% X 6% B = 5% l —5% X 6% I
.997 B = .047 I
PROOF
Let I = $100.00
.997 B = .047 X $100 = $4.70
B = $4.70
$4.7141
.997
T = 6% (I — B )
= 6% ($100.00 — $4.7141)
= 6% X $95.2859 = $5.717154
B = 5% (I — T )
= 5% ($100.00 —$5.717154)
= 5% X $94.282846 = $4.7141423
Net Income (I) $100.00
Taxes (T) 5.717
94.283
.05
B = 5% (I — T ) $4.71415
Formula for computation of bonus based
on net income remaining after deduction of
both income tax and the bonus.
$100.00 = income before deducting either
tax or bonus
X = amount of bonus
Y = amount of taxes
Y = .06 (100 — X . )
X = .05 (100 — X — Y )
100Y = 600 — 6X
100X = 500 — 5X — 5Y
100Y = 600 — 6X
100Y= 10,000 — 2100X
0 = 9400 — 2094X
94 (100)
X = 4.489016
2094
Y = 6 — .06X = 6 — (.06 X 4.489016) =
6 — .26934096 = 5.73065904
PROOF
Original Income $100.00
Bonus 4.49
95.51
Tax @ 6% 5.73
89.78
Bonus @ 5% 4.489