SOLUTION: I get so lost in these long problems please help The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD (Certificate of

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Question 83291: I get so lost in these long problems please help
The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD (Certificate of Deposit) is given by

A is the amount of returned.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the compound period.
t is the number of years.
Carry all calculations to 6 decimals on all assignments then round the answer to the nearest cent.
Suppose you deposit $10,000 for 2 years at a rate of 10%.
a) Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the hundredth's place.
Answer:
a) continued
Show work in this space. Use ^ to indicate the power.



b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
Answer:
Show work in this space



c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.
Answer:
Show work in this space



d) Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.
Answer:
Show work in this space

e) What observation can you make about the size of the increase in your return as your compounding increases more frequently?
Answer:



f) If a bank compounds continuously, then the formula takes a simpler, that is

where e is a constant and equals approximately 2.7183.
Calculate A with continuous compounding. Round your answer to the hundredth's place.
Answer:
Show work in this space



g) Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t). Round your answer to the hundredth's place.
Answer:
Show work in this space


h) A commonly asked question is, “How long will it take to double my money?” At 10% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place.
Answer:
Show work in this space



Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!

Using this formula A=P%281%2Br%2Fn%29%5E%28nt%29 calculate the following:

a)Calculate the return (A) if the bank compounds annually (n = 1)
A=10000%281%2B0.10%2F1%29%5E%281%2A2%29 Start with the given expression
A=10000%281%2B0.1%29%5E%282%29 Divide 0.10 by 1 and multiply the exponents 1 and 2
A=10000%281.1%29%5E%282%29 Add
A=10000%281.21%29 Raise 1.1 to 2
A=12100 Multiply

So the return is $12,100


b)Calculate the return (A) if the bank compounds quarterly (n = 4)
A=10000%281%2B0.10%2F4%29%5E%284%2A2%29 Start with the given expression
A=10000%281%2B0.025%29%5E%288%29 Divide 0.10 by 4 and multiply the exponents 4 and 2
A=10000%281.025%29%5E%288%29 Add
A=10000%281.21840289750992%29 Raise 1.025 to 8
A=12184.0289750992 Multiply

So the return is $12184.03


c)Calculate the return (A) if the bank compounds monthly(n = 12)
A=10000%281%2B0.10%2F12%29%5E%2812%2A2%29 Start with the given expression
A=10000%281%2B0.00833333333333333%29%5E%2824%29 Divide 0.10 by 12 and multiply the exponents 12 and 2
A=10000%281.00833333333333%29%5E%2824%29 Add
A=10000%281.22039096137556%29 Raise 1.00833333333333 to 24
A=12203.9096137556 Multiply

So the return is $12203.91

d)Calculate the return (A) if the bank compounds daily (n = 365)
A=10000%281%2B0.10%2F365%29%5E%28365%2A2%29 Start with the given expression
A=10000%281%2B0.000273972602739726%29%5E%28730%29 Divide 0.10 by 365 and multiply the exponents 365 and 2
A=10000%281.00027397260274%29%5E%28730%29 Add
A=10000%281.22136930163979%29 Raise 1.00027397260274 to 730
A=12213.6930163979 Multiply

So the return is $12213.69


e) What observation can you make about the size of the increase in your return as your compounding increases more frequently?

As the compounding frequency increases, the return slowly approaches some finite number (which in this case appears to be about $12213.69). Think about it, banks wouldn't be too fond of shelling out an infinite amount of cash.


f)Calculate A with continuous compounding
Using the contiuous compounding formula A=Pe%5E%28rt%29 where e is the constant 2.7183 and letting r=0.1, P=10,000, and t=2 we get

A=10000%282.7183%29%5E%280.1%2A2%29 Start with the given equation

A=10000%282.7183%29%5E%280.2%29 Multiply 0.1 and 2

A=10000%281.22140439115573%29 Raise 2.7183 to 0.2

A=12214.0439115572 Multiply

So using continuous compounding interest we get a return of $12,214.04 (which is real close to what we got from a daily compounding frequency)


g)Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t)

15000=10000e%5E%280.1t%29

15000%2F10000=e%5E%280.1t%29 Divide both sides by 10,000

1.5=e%5E%280.1t%29
ln%281.5%29=0.1t Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.

ln%281.5%29%2F0.1=t Divide both sides by 0.1

So we get

t=0.4054%2F0.1=4.054
t=4.054

So it will take about 4 years to generate $15,000


h) A commonly asked question is, “How long will it take to double my money?” At 10% interest rate and continuous compounding, what is the answer?

Since we want to double our money, let A=2*10,000. So A=20,000. Now solve for t:
20000=10000e%5E%280.1t%29

20000%2F10000=e%5E%280.1t%29 Divide both sides by 10,000

2=e%5E%280.1t%29
ln%282%29=0.1t Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.

ln%282%29%2F0.1=t Divide both sides by 0.1

So we get

t=0.69314%2F0.1=6.9314
t=6.9314

So it will take about 7 years to double your money.