Ben's Money Tree
Consider this statement taken from the will of Benjamin Franklin:
The said sum of one thousand pounds sterling, if accepted by the inhabitants of the town of Boston, shall be managed under the direction of the selectmen... who are to let out the sum upon interest, at five per cent, per annum, to such young married artificers, under the age of twenty-five years, as have served an apprenticeship in the said town, and faithfully fulfilled the duties required in their indentures, so as to obtain a good moral character from at least two respectable citizens, who are willing to become their sureties, in a bond with the applicants, for the repayment of the moneys so lent, with interest, according to the terms hereinafter prescribed; all which bonds... or the value thereof in current gold coin; and the managers shall keep a bound book or books, wherein shall be entered the names of those who shall apply for and receive the benefits of this institution, and of their sureties, together with the sums lent, the dates, and other necessary and proper records respecting the business and concerns of this institution. And as these loans are intended to assist young married artificers in setting up their business, they are to be proportioned by the discretion of the managers, so as not to exceed sixty pounds sterling to one person, nor to be less than fifteen pounds; and if the number of appliers so entitled should be so large as that the sum will not suffice to afford to each as much as might otherwise not be improper, the proportion to each shall be diminished so as to afford to every one some assistance. These aids may, therefore, be small at first, but, as the capital increases by the accumulated interest, they will be more ample. And in order to serve as many as possible in their turn, as well as to make the repayment of the principal borrowed more easy, each borrower shall be obliged to pay, with the yearly interest, one tenth part of the principal and interest, so paid in, shall be again let out to fresh borrowers.
Question 1: This account still exists today. Since the 1000 pounds (equivalent to $4600 at that time) was loaned out at 5% interest compounded annually, what should be its current value?
Question 2: Repeat Question 1, except suppose Benjamin Franklin had been wise enough to make the interest rate of 5% be compounded continuously?
Hint: What are the key formulas for determining P...as Po, r, and time t are all known?
Solution Commentary: And crank out your answer......?