SOLUTION: I just want to know if someone could check this and see if it is right and help me on the rest. Not from a textbook. Thank you 3) The formula for calculating the amount of

Question 82927: I just want to know if someone could check this and see if it is right and help me on the rest. Not from a textbook. Thank you

3) 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: $2,200 is the return
Show work in this space. Use ^ to indicate the power. A= 10,000(1 + .10/1)^1 * 2

b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
Answer: $8,200
Show work in this space A = 10,000.00(1 +.10/4)^4*2

c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.
Answer: $2,419.20
Show work in this space
10,000.00(1+.10/12)^12*2

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(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation

Plug in p=10000, r=0.1

a)
Lets calculate the return if the bank compounds annually
Let n=1 and plug it into
Start with the given expression
Divide 0.1 by 1 to get 0.1
Multiply the exponents 1 and 2 to get 2
Add 1 and 0.1 to get 1.1
Raise 1.1 to 2 to get 1.21
Multiply 10000 and 1.21 to get 12100
So our return is $12100
b)
Lets calculate the return if the bank compounds quarterly
Let n=4 and plug it into
Start with the given expression
Divide 0.1 by 4 to get 0.025
Multiply the exponents 4 and 2 to get 8
Add 1 and 0.025 to get 1.025
Raise 1.025 to 8 to get 1.21840289750992
Multiply 10000 and 1.21840289750992 to get 12184.0289750992
So our return is $12184.03
c)
Lets calculate the return if the bank compounds monthly
Let n=12 and plug it into
Start with the given expression
Divide 0.1 by 12 to get 0.00833333333333333
Multiply the exponents 12 and 2 to get 24
Add 1 and 0.00833333333333333 to get 1.00833333333333
Raise 1.00833333333333 to 24 to get 1.22039096137556
Multiply 10000 and 1.22039096137556 to get 12203.9096137556
So our return is $12203.91
d)
Lets calculate the return if the bank compounds daily
Let n=365 and plug it into
Start with the given expression
Divide 0.1 by 365 to get 0.000273972602739726
Multiply the exponents 365 and 2 to get 730
Add 1 and 0.000273972602739726 to get 1.00027397260274
Raise 1.00027397260274 to 730 to get 1.22136930163979
Multiply 10000 and 1.22136930163979 to get 12213.6930163979
So our 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 where e is the constant 2.7183 and letting r=0.1, P=10,000, and t=2 we get

Start with the given equation

Multiply 0.1 and 2

Raise 2.7183 to 0.2

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)

Divide both sides by 10,000

Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.

Divide both sides by 0.1

So we get

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:

Divide both sides by 10,000

Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.

Divide both sides by 0.1

So we get

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