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Question 1133490: If $100 invested at time 0 accumulates to $172 at time 3, find the accumulated value at time 10 of $100 invested at time 5.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you didn't specify whether it was compound interest or simple interest.
if it was compound interest, the formula is f = p * (1 + r) ^ n
n = time period 3 minus time period 0 = 3
f = 172
p = 100
formula becomes 172 = 100 * (1 + r) ^ 3
divide both sides of the equation by 100 and take the third root of both sides of the equation to get:
(172/100) ^ (1/3) = 1 + r
solve for r to get r = (172/100) ^ (1/3) - 1.
this makes r = .198145283.
that's the interest rate per time period.
f = 100 * (1.198145283) ^ 3 becomes f = 172.
the interest rate per time period is good.
time period 10 minus time period 5 = 5
100 invested in time period 5 at the same rate for 5 time periods makes the formula equal to:
f = 100 * (1.198145283) ^ 5 which makes f equal to 246.9149645.
the accumulated value at time period 10 is 246.9149645.
the formula for simple interest is:
i = p * r * n
i is the interest
p is the principal
r is the interest rate per time period.
n is the number of time period.
the future value of the investment is f = p + i.
since i = p * r * n, you get f = p + p * r * n.
you can factor out the p to get f = p * (1 + r * n)
when p = 100 and f = 172 and n = 3, the formula becomes:
172 = 100 * (1 + r * 3)
divide both sides of the equation by 100 and subtract 1 from both sides of the equation and then divide both sides of the equation by 3 to get:
(172/100 - 1) / 3 = r
solve for r to get r = .24
the simple interest rate is .24 per time period.
the original equation for f is f = p + p * r * n
that becomes f = 100 + 100 * .24 * 3 which becomes f = 172.
when n = 5, the formula becomes f = 100 + 100 * .24 * 5 which becomes f = 220.
with compound interest formula, the interest rate per time period is .198145283.
in 3 years, 100 grows to 172.
in 5 years, 100 grows to 246.9149645.
with simple interest formula, the interest rate per time period is .24.
in 3 years, 100 grows to 172.
in 5 years, 100 grows to 220.
compound interest formula graph is shown below followed by simple interest formula graph.
equation for compound interest graph is y = 100 * 1.198145283 ^ x
equation for simple interest graph is y = 100 + 100 * .24 * x
here's the graphs.
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