9514 1404 393
Answer:
4%
Step-by-step explanation:
The multiplier for computing simple interest over a 3-year period is ...
I = Prt . . . . . interest on principal P at rate r for t years
I/P = rt = 3r
The multiplier for computing compound interest over a 3-year period is ...
A = P(1 +r)^t . . . . amount with interest
A/P -1 = (1 +r)^3 -1 . . . . multiplier for computing interest only
Then the multiplier for the difference in amounts of interest is ...
((1 +r)^3 -1) -3r = (1 +3r +3r^2 +r^3) -1 -3r = r^2(3 +r)
This means we need to solve the cubic ...
r^2(3 +r) = 2.432/500
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This can be solved a variety of ways. Perhaps my favorite is a graphical solution. For that, we would rewrite the equation to the form ...
f(x) = x^2(3 +x) -0.004864
so the solution is found where f(x) = 0. The graph shows the solution to be ...
x = 0.04
The rate of interest is 0.04 = 4%.
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Additional comment
Using the above f(x) to develop a Newton's Method iterator, we find it to be ...
x' = (x²(2x +3) +0.004864)/(3x(x +2))
This needs a starting value, so we need an initial clue as to the approximate interest rate.
That can be offered by rearranging the cubic we first developed:
r = √(0.004864/(r +3))
Starting with r=0, this gives r ≈ 0.0402. At this point, we can either use this equation as an iterator, or use the Newton's method iterator above. Both will give an interest rate accurate to 4 or 5 decimal places in one iteration.
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You may be aware from the development above that the polynomial that must be solved will have a degree equal to the number of years over which interest is computed. That is, solving the same question for a 10-year period would involve finding the root of a 10th-degree polynomial.
