SOLUTION: Frank invests $3500 dollars into an account earning interest compounded quarterly. After 10 years there is $5050 in the account. What was the interest rate?

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Question 1198963: Frank invests $3500 dollars into an account earning interest compounded quarterly. After 10 years there is $5050 in the account. What was the interest rate?
Found 3 solutions by MathLover1, josgarithmetic, math_tutor2020:
Answer by MathLover1(20849) About Me  (Show Source):
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using the following formula:
A=P%281%2Br%2Fn%29%5E%28n%2At%29
given:
A=5050
P=3500
t=10
n=4

5050=3500%281%2Br%2F4%29%5E%284%2A10%29
5050%2F3500=%281%2Br%2F4%29%5E40
%281%2Br%2F4%29%5E40=101%2F70
log%28%281%2Br%2F4%29%5E40%29=log%28101%2F70%29
40log%28%281%2Br%2F4%29%29=log%28101%2F70%29
log%28%281%2Br%2F4%29%29=log%28101%2F70%29%2F40

log%28r%2F4+%2B+1%29+=+0.009165631869797533
r≈0.036831
r=3.6831%

Answer by josgarithmetic(39617) About Me  (Show Source):
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10 years as quarterly periods means 40 compounding periods.
r for yearly rate means r%2F4 for quarterly rate.

3500%281%2Br%2F4%29%5E40=5050
.
.
1%2Br%2F4=1.00920776
r%2F4=0.00920776
highlight%28r=0.0368%29
or 3.68%

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Compound interest formula
A = P*(1+r/n)^(n*t)

Variables:
A = final account value after t years
P = deposit or starting value
r = annual interest rate in decimal form
n = compounding frequency
t = number of years

In this case:
A = 5050
P = 3500
r = unknown
n = 4
t = 10

Let's solve for the variable r.
A = P*(1+r/n)^(n*t)
5050 = 3500*(1+r/4)^(4*10)
5050/3500 = (1+r/4)^(40)
1.442857143 = (1+r/4)^(40)
(1+r/4)^(40) = 1.442857143
1+r/4 = (1.442857143)^(1/40)
1+r/4 = 1.009207765
r/4 = 1.009207765 - 1
r/4 = 0.009207765
r = 4*0.009207765
r = 0.03683106
r = 0.0368

r*(100%) = 0.0368*100% = 3.68%
In other words, we move the decimal point two spots to the right to go from 0.0368 to 3.68%

The annual interest rate is roughly 3.68%

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To verify the answer, we can use a TVM calculator such as this
https://www.geogebra.org/m/mvv2nus2
A similar feature is found on any TI83 or TI84 calculator.
TVM = time value of money

The input values are
N = 10
I% = left blank
PV = -3500
PMT = 0
FV = 5050
P/Y = 1
C/Y = 4

N represents the number of periods. In this case it's the number of years.
PV is negative to represent a cash outflow.
PMT represents the periodic payment. We type zero here to indicate we don't have periodic payments; rather we have a one-time deposit instead.
C/Y stands for "compoundings per year"

After you input those items into the proper boxes, press the "solve for I%" button to have 3.68 show up in that box.

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Answer: 3.68% (approximate)

Another similar finance question that involves TVM:
https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1199059.html