Overlapping Clock Hands and Tax Free Gifts
Tax Free Money Suppose that you wanted to give someone a tax free gift of $1,000. How much do you have to give them if the tax rate is 20% ? A quick calculation would add 20% of $1,000 or $200 to the amount, and give $1,200. But 20% of $1,200 is $240, so the recipient gets only $960 instead of $1,000 after paying the tax. The recipient gets $40 less than intended, so adding $40 for a total of $1,240 would seem to be the remedy. But the tax on this is $248 leaving the recipient $8 short. This is an iterative problem which requires repeating the above procedure many more times to calculate the required total so that the recipient nets $1,000 after taxes. The sequence, a geometric infinite series, is as follows: where S is the required sum, and n approaches infinity. The sum increases as shown below: The iterations in [2] ending with 0.32 totals to 1249.92. More iterations show that the required total is approaching 1250. There is an easier way. Factor out 1000 in [1] gives: Multip...