A rich man has 1 million dollars in the bank earning 7% interest. He plans to give away $100,000 at the end of the year and to increase his gifting by 10% per year thereafter. How long will the million dollars last?
To find out how long the million dollars will last, we need to calculate the amount of money the rich man will have each year after giving away $100,000 and adjusting for the increasing gifting amount.
First, let's determine the amount of interest the rich man earns each year. The interest is calculated using the formula:
Interest = Principal × Rate
Where the principal is $1,000,000 and the rate is 7%. We can calculate the interest as:
Interest = $1,000,000 × 7% = $70,000
So, in the first year, the rich man will have an additional $70,000 from the interest.
Next, let's calculate the amount he will give away in the second year. It will be $100,000 increased by 10%. We can express this as:
Gift_2 = Gift_1 + (10% × Gift_1)
Where Gift_1 is $100,000. Plugging in the values:
Gift_2 = $100,000 + (10% × $100,000) = $100,000 + $10,000 = $110,000
Each subsequent year, the gifting amount will increase by 10%.
Now, let's calculate the amount of money the rich man will have left after giving away $100,000 and accounting for the interest earned. We can express this as a recursive formula:
Money_left_n = Money_left_(n-1) + Interest - Gift_(n-1)
Where Money_left_n represents the amount of money the rich man has after n years, and Gift_(n-1) represents the gifting amount for that year.
Using this formula, let's calculate the amount of money left each year until it reaches zero:
Year 1:
Money_left_1 = $1,000,000 + $70,000 - $100,000 = $970,000
Year 2:
Money_left_2 = $970,000 + $70,000 - $110,000 = $930,000
Year 3:
Money_left_3 = $930,000 + $70,000 - ($110,000 × 1.10) = $877,000
Year 4:
Money_left_4 = $877,000 + $70,000 - ($110,000 × 1.21) = $810,700
Year 5:
Money_left_5 = $810,700 + $70,000 - ($110,000 × 1.331) = $730,360.30
Year 6:
Money_left_6 = $730,360.30 + $70,000 - ($110,000 × 1.4641) = $636,960.43
Year 7:
Money_left_7 = $636,960.43 + $70,000 - ($110,000 × 1.61051) = $530,511.20
Year 8:
Money_left_8 = $530,511.20 + $70,000 - ($110,000 × 1.77156) = $410,621.20
Year 9:
Money_left_9 = $410,621.20 + $70,000 - ($110,000 × 1.94871) = $277,530.60
Year 10:
Money_left_10 = $277,530.60 + $70,000 - ($110,000 × 2.14358) = $131,488.82
Based on these calculations, the million dollars will last approximately 10 years before it runs out, assuming the given interest rate and gifting pattern.