# Compound Interest

1. Oct 17, 2017

### zak100

1. The problem statement, all variables and given/known data

Amy deposited $1000 into an account that earns 8% annual interest compounded every 6 months. Bod deposited$1000 into an account that earns 8% annual interest compounded quarterly. If neither Amy nor Bob makes any additional deposits or withdrawals in 6 months, how much more money will Bob have in his account than Amy?

2. Relevant equations

A = P(1+r/n)^nt

3. The attempt at a solution
For Ammy

P=1000, t=1( annual), n=2 (compounded 6 months), r=8%=0.08
A= 1000(1+0.08/2)^2= 1000(1.04)^2=1.0816= 1081.6

Kindly tell why is the solution wrong?

Zulfi.

2. Oct 17, 2017

### phyzguy

You calculated how much money Amy will have after 1 year. The problem says, "in 6 months, how much more money will Bob have in his account than Amy?"

3. Oct 17, 2017

### zak100

Hi,
I am to find correct value for Ammy:

P=1000, t=1( annual), n=1 (compounded 6 months and we are calculating only for the 1st period not the whole year), r=8%=0.08
A= 1000(1+0.08/2)^2= 1000(1.04)^1=1040
This is correct.
For Bob:

P=1000, n=2(compounded quarterly , so 4 times but we have to calculate for 6 months so n=2, r=0.08, t=1

Now A = P (1 + r/n)^ (nt)

A= 1000(1 + 0.08/2)^(2*1)

Zulfi.

4. Oct 17, 2017

### phyzguy

Amy looks correct now. For Bob, it should be (1 + 0.08/4)^2.

5. Oct 17, 2017

### SammyS

Staff Emeritus
In your relevant equation section, What value should you use for t in both cases ?

A = P(1+r/n)nt
.

Last edited: Oct 20, 2017
6. Oct 17, 2017

### zak100

Hi,
I have used t=1 in both the cases, because it says "annual interest". I cant understand why n=4 in Bob case. It says quarterly so its 4 ( periods of interest of 3 months) but for 6 months we should have n=2 because there are 2 periods of 3 months.

Zulfi.

7. Oct 18, 2017

### phyzguy

If you get 8% interest per year, you only get 2% in 3 months, not 4%.

8. Oct 20, 2017

### SammyS

Staff Emeritus
t is the number of years. The question asks ".. in 6 months, how much more money will Bob have in his account than Amy?"

I'm pretty sure that 6 months is 1/2 year rather than i year, so t = 0.5, in both cases.

9. Oct 21, 2017

### zak100

Hi,
Thanks. I have solved this problem.
For Amy:

A = the amount of money that Ammy has after 6 months, because the problem asks how much he has after 6 months.=?
P = the original amount of money that Amy invested (principal) -1000
r = the annual interest rate = ?? (that is a 12 month rate) =0.08
n = the number of times that interest is compounded per year = 2
t = the number of years the money is invested =.5A = P (1 + r/n)^ (nt)
A= 1000(1+0.08/2)^(2 *1/2)
A= 1000(1+0.04) ^1 = 1000 (1.04) = 1040

For Bob:
A = the amount of money that Bob has after 6 months, because the problem asks how much he has after 6 months.=?
P = the original amount of money that Amy invested (principal) -1000
r = the annual interest rate = ?? (that is a 12 month rate) =0.08
n = the number of times that interest is compounded per year = 4
t = the number of years the money is invested =.5
A = P (1 + r/n)^ (nt)
A= 1000(1+0.08/4)^(4*1/2) = 1000(1+0.02)^2 = 1000(1.02)^2 = 1.0404=1040.4
I think I am right this time.

Now difference =A=1040.4-1040=0.4

Zulfi.
Zulfi.