   SEARCH HOME 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.