 |
| ||
| |||
|
| Math Central | Quandaries & Queries |
|
Question from April, a student: Taxylvania has a tax code that rewards charitable giving. If a person gives p% of his income to charity, that person pays (35-1.8p)% tax on the remaining money. For example, if a person gives 10% of his income to charity, he pays 17 % tax on the remaining money. If a person gives 19.44% of his income to charity, he pays no tax on the remaining money. A person does not receive a tax refund if he gives more than 19.44% of his income to charity. Count Taxula earns $27,000. What percentage of his income should he give to charity to maximize the money he has after taxes and charitable giving? |
Hi April,
If Count Taxula gives p% of his income to charity he gives p/100 × $27,000 which leaves him with
$27,000 - p/100 × $27,000 = (1 - p/100) × $27,000
He has to pay a tax of (35 - 1.8p)% on this remaining amount, so his tax is
(35 - 1.8p)/100 × (1 - p/100) × $27,000
Thus after charitable giving and taxes he is left with
(1 - p/100) × $27,000 - (35 - 1.8p)/100 × (1 - p/100) × $27,000
= (1 - p/100) × $27,000 × [1 - (35 - 1.8p)/100]
= $2.7 × (100 - p)(65 + 1.8p)
This is the expression you need to maximize for 0% ≤ p ≤ 19.44%
I hope this helps,
Harley
| ||
| ||
| ||
 |
 |
|
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.