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1. Quadrupling in 24 years is doubling in 12 years, so the rate is 6%. The exact value is 5.9463094%.
2. The rule is exact at (slightly less than) 8%, but no whole number n corresponds to that rate of interest.
3. Let n be the number of periods at interest rate i (not in percent) per period. Since there is a doubling, (1+i)^n=2, or taking natural logs, n*ln(1+i)=ln(2). Replace ln(1+i) by (ln(1.08)/.08)*i, the secant line from (0,0) to (.08,ln(1.08)). This yields n*i=.720517467, so that rounding off and multiplying both sides by 100, we get n*(100*i)=72. Denoting by n' the approximate value of n gotten using the rule, we have n'/n ~= 1.04*(ln(1+i)/i), and for i=.2 (20% interest) this ratio is .95, while for i=.01 (1% interest) it is 1.03. |
