Question

Assignment: Compound Interest Investigation

Tyler has earned money by helping out on a neighbor's farm and now wants to put his earnings in a savings account. He is a bit confused on the different interest options available and how each will impact the amount he has after several years. Help Tyler better understand by showing him how his money will increase in an account that uses simple interest and one that uses compound interest over a specified period of time.

1. a. If Tyler deposits $1500 of the $3200 he has earned in an account that pays 4% interest compounded annually, how much will he have in his account after 5 years?

b. If Tyler deposits $1500 of the $3200 he has earned in an account that pays 4% simple interest, how much will he have after 5 years? Show all work.

c. Which account would yield a greater amount? What is the difference between the two amounts?

2. a. If Tyler deposits $2000 of the $3200 he has earned in an account that pays 8% interest compounded quarterly, how much will he have in his account after 1 year?

b. If Tyler deposits $2000 of the $3200 he has earned in an account that pays 8% simple interest, how much will he have after 1 year? Show all work.

c. Which account would yield a greater amount? What is the difference between the two amounts?


3. Tyler decides to deposit his earnings of $3200 in a savings account. Bank A offers an account with a simple interest rate of 3.5%. Bank B offers an interest rate of 3.4% compounded annually. After 3 years, in which bank will Tyler have the greatest total? Show all work.

a. Find the interest earned for Bank A.

b. Find the total amount for Bank B.

c. Which bank should Tyler choose if he wants to earn the greater amount? Explain your reasoning.

d. Which account would yield a greater amount? What is the difference between the two amounts?

Answer 1

To help Tyler better understand how his money will increase in an account that uses simple interest and one that uses compound interest, we are going to use two formulas: a simple interest formula for the accounts that use simple interest, and a compound interest formula for the accounts that use compound interest.
- Simple interest formula:
`A=P(1+rt)`
where:

`A` is the final investment value

`P` is the initial investment

`r` is the interest rate in decimal form

`t` is number of years
- Compound interest formula:
`A=P(1+ (r)/(n) )^(nt)`
where:

`A` is the final investment value

`P` is the initial investment

`r` is the interest rate in decimal form

`t` is he number of years

`n` is the number of times the interest is compounded per year

1.
a. This is a compound interest account, so we are going to use our compound interest formula. We now that
`P=1500`,
`t=5`, and since the interest is compounded annually (1 time a year),
`n=1`. To find the interest rate in decimal form, we are going to divide it by 100%:
`r= (4)/(100) =0.04`. Now that we have all the values lets replace them in our compound interest formula:

`A=1500(1+ (0.04)/(1)) ^((1)(5))`

`A=1824.98`
We can conclude that after 5 years he will have $1824.98 in this account.
b. Here we will use our simple interest formula. We know that
`P=1500`,
`t=5`, and
`r= (4)/(100) =0.04`. Lets replace those values in our simple interest formula:

`A=1500(1+(0.04)(5))`

`A=1800`
We can conclude that after 5 years he will have $1800 in this account.
c. The compound interest account from point a will yield more money than the simple account one from point b. The difference between the tow amounts is
`1824.98-1800=24.98`

2.
a. Here we are going to use our compound interest formula. We know that
`P=2000`,
`t=1` and
`r= (8)/(100) =0.08`. We also know that the interest is compounded Quaternary (4 times per year), so
`n=4`. Now that we have all our values lets replace them into our formula:

`A=2000(1+ (0.08)/(4) )^((4)(1))`

`A=2164.86`
We can conclude that after 1 year he will have $2164.86 in this account.
b. Here we are going to use our simple interest formula. We know that
`P=2000`,
`t=1`, and
`r= (8)/(100) =0.08`. Once again, lets replace those values in our formula:

`A=2000(1+(0.08)(1))`

`A=2160`
We can conclude that after 1 year he will have $2160 in this account.
c. The compound interest account from point a will yield more money than the simple account one from point b. The difference between the tow amounts is
`2164.86-2160=4.86`

3.
a. Since Bank A offers an account with a simple interest, we are going to use our simple interest formula. From the question we know that
`P=3200`,
`t=3`, and
`r= (3.5)/(100) =0.035`. Now we can replace those values into our formula to get:

`A=3200(1+(0.035)(3))`

`A=3536`
Now, to find the interest earned for Bank A we are going to subtract
`P` from
`A`

`InterestEarned=3536-3200=336`
We can conclude that the interest earned for Bank A is $336
b. Since Bank B offers an account with a compound interest, we are going to use our compound interest formula. We know that
`P=3200`,
`t=3`,
`r= (3.4)/(100) =0.034`, and since the interest is compounded annually (1 time a year),
`n=1`. Now that we have all the values, lets replace them in our formula to get:

`A=3200(1+ (0.034)/(1) )^((1)(3))`

`A=3537.62`
Now, to find the interest earned for Bank A we are going to subtract
`P` from
`A`:

`InterestEarned=3537.62-3200=337.62`
We can conclude that the interest earned for Bank B is $337.62
c. Even tough the interest returns between the tow Banks are very similar, Bank B offers a slightly better interest over a period of time, which can make a big difference in the long run. If Tyler wants the earn more money, he definitively should deposit his money in Bank B.
d. The compound interest account from Bank B will yield more money than the simple account one from Bank A The difference between the tow amounts is
`3537.62-3536=1.62`