Questions on Word Problems: Money, Business and Interest answered by real tutors!

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Question 152528This question is from textbook
: Please help not really understanding this problem: The formula for calculating the amount of money returned for an initial deposit into a bank account or CD is given by A=P(1+r/n)^nt
A is the amount of the return.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the number of compound periods in one year.
t is the number of years.
Carry all calculations to six decimals on each intermediate step, then round the final answer to the nearest cent.
Suppose you deposit $3000 for 9 years at a rate of 6%
a)Calculate the return(A) if the bank compounds annually (n=1). Round your answer to the hundredth's place.
b)Calculate the return (A) if the bank compounds quarterly (n=4). Round your answer to the hundredth's place.
c) Does compounding annually or quarterly yield more interest? Why?
d)If a bank compounds continuously, then the formula userdf is A=Pe^rt where e is a constant and equals approximately 2.7183. Calculate A with continuous compounding. Round your answer to the hundredth's place.
e) How long will it take to double my money? At 6% interest rate and continuous compounding , what is the answer? Round your answer to the hundredth's place.
Thank you so much for your help in advance.This question is from textbook
: Please help not really understanding this problem: The formula for calculating the amount of money returned for an initial deposit into a bank account or CD is given by A=P(1+r/n)^nt
A is the amount of the return.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the number of compound periods in one year.
t is the number of years.
Carry all calculations to six decimals on each intermediate step, then round the final answer to the nearest cent.
Suppose you deposit $3000 for 9 years at a rate of 6%
a)Calculate the return(A) if the bank compounds annually (n=1). Round your answer to the hundredth's place.
b)Calculate the return (A) if the bank compounds quarterly (n=4). Round your answer to the hundredth's place.
c) Does compounding annually or quarterly yield more interest? Why?
d)If a bank compounds continuously, then the formula userdf is A=Pe^rt where e is a constant and equals approximately 2.7183. Calculate A with continuous compounding. Round your answer to the hundredth's place.
e) How long will it take to double my money? At 6% interest rate and continuous compounding , what is the answer? Round your answer to the hundredth's place.
Thank you so much for your help in advance.
Answer by mducky2(55) About Me  (Show Source):
You can put this solution on YOUR website!
Part A: We can just plug in the numbers:
The principal amount (P) is 3000, since that is what was originally deposited.
The rate (r) is 0.06 because 6% means 6/100.
The number (n) that it is compounded is 1, since annually means only once a year.
The time (t) is 9 years.

The formula is much easier to deal with when we first plug in n=1, so let's start with that:
A = P(1+(r/n))^nt
A = P(1+(r/1))^(1t)
A = P(1+r)^t

This is also the general formula for the return on any deposit compounded annually. Now we can plug in the specific numbers:
A = 3000(1+0.06)^9
A = 3000(1.06^9)
A = 3000(1.689478)
A = 5068.436877

The return is $5068.44.


Part B: We can just plug in the numbers. P, r, and t are the same, but now n changes from 1 to 4:
A = P(1+(r/n))^nt
A = P(1+(r/4))^(4t)

This is also the general formula for the return on any deposit compounded quarterly. Now let's plug in the numbers:
A = 3000(1+(0.06/4))^(4*9)
A = 3000(1+0.015)^36
A = 3000(1.015)^36
A = 3000(1.709139)
A = 5127.418614

The return is $5127.42


Part C: Compounding quarterly yields more interest. This is because when we do it once a year, it only multiplies the whole thing once by 1.06. When we do it four times a year, it multiplies it by 1.015^4, which is 1.06136355, which is actually more than 1.06.


Part D: Now we will use a different formula entirely.
A = Pe^rt

We can still plug in the same numbers for P, r, and t.
A = 3000e^(9*0.06)
A = 3000e^(.54)
A = 3000(1.7160069)
A = 5148.020586

The return is $5148.02


Part E: In order to find out how much it will take to double the money, we start with the equation:
A = 2P = Pe^rt

The variables P and r will be the same, but we no longer know how much time it will take.
2(3000) = 3000e^(0.06*t)
6000 = 3000e^(0.06*t)
6000/3000 = (3000e^(0.06*t))/3000
2 = e^(0.06*t)

It looks like we need to use logarithms to solve this problem. The natural log of 2 will equal 0.06*t:
ln (2) = 0.06*t
(ln (2))/0.06 = t
t = 11.552453

It should take 11.55 years for the deposit to double.