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Question 1171535: April loaned her friend 750 to buy guitar he agreed to pay back the money with 5% annual interest at the end of 6 month how much did her friend give her at the end of 6 months
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if the interest rate was compounded semi-annually, then 5% / 2 = 2.5% and he would have paid her back 750 * 1.025 = 768.75
the effective annual rate, in that case, would have been 1.025^2 = 1.050625 - 1 = .050625 * 100 = 5.0625%.
if the interest rate was not compounded semi-annually, then he would have paid her back 750 * 1.05^(1/2) = 768.5213074.
this is slightly less than 768.75.
the difference is in the compounding.
with annual compounding, the annual interest rate is divided by 1 and the number of years is multiplied by 1.
with semi-annual compounding, the annual interest rate is divided by 2 and the number of years is multiplied by 2.
with quarterly compounding, the annual interest rate is divided by 4 and the number of years is multiplied by 4.
with monthly compounding, the annual interest rate is divided by 12 and the number of years is multiplied by 12.
with daily compounding, the annual interest rate is divided by 365 and the number of years is multiplied by 365.
the higher the number of compounding periods per year, the higher the effective interest rate per year.
the highest number of compounding periods per year it can go is continuous compounding.
that's a different formula.
that formula is f = p * e ^ (r * t)
f is the future value
p is the present value
r is the interest rate per time period.
t is the number of time periods.
with continuous compounding, the rate is usually left as the annual rate and the number of years is left as the number of year.
the formula will give you the same effective interest rate regardless of the time periods you used.
with discrete compounding, the formula is:
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.
all of the above assumes compound interest.
if you are taking about simple interest, then a different formula applies.
that formula is:
f = p + p * r * n which can also be shown as:
f = p * (1 + r * n)
in your case, simple interest formula would give you:
f = 750 * (1 + .05 * .5) = 768.75.
while you get the same answer as if the interest rate was compounded semi-annually, it is not the same.
to see the difference, assume the loan was for 20 years.
at simple interest, he would have had to pay 750 * (1 + .05 * 20) = 1500.
at 5% compounded semi-annually, he would have had to pay 750 * (1 + .025)^40 = 2013.7978879.
the difference gets larger, the longer the term of the loan.
the following table shows you the difference as the length of the loan increases.
for your problem you would be looking at nyrs = .5.
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