Math
posted by Cathy
Can someone steer me in the right direction? Here's the question.
Envision that you have served as business manager of Media World for over 2 years. You have noticed that for the last 12 months the business has regularly had cash assets of $20,000 or more at the end of each month. You have found a 6month certificate of deposit that pays 6% compounded monthly. To obtain this rate of interest, you must invest a minimum of $2,000. You have also found a high interest savings account that pays 3% compounded daily. Based on the cash position of the business at this time, assume that you decide to invest $4,000
1. Assume that you will invest the full amount in a certificate of deposit.
a. What would be the future value of the CD at the end of the investment term?
b. How much interest would the investment earn for the period?
c. What would be the effective rate of the investment?

jim
One reason I'm leery of this question is "6% compounded monthly". If that's supposed to mean 6% per month, which is what it seems to mean, it's a crazy interest rate, over 100% p.a. (3% compounded daily is even crazier! Waay crazier!)
a.
Anyway, in general, if you have x% interest compounded, you multiply by
(100+x)/100 at every interval, which means that if you have y intervals, your original money is multiplied by
((100+x)/100)^y
6% compounded at each of 6 months would therefore multiply your original capital by
1.06^6
b. Given that you know the final value, subtract your original capital, and what remains is interest.
c. I'm not sure what they mean by "effective rate". If they mean the equivalent % p.a. the answer will be about 101.2% 
Cathy
Thanks for answering jim. So, for a. that would be $4000 + 6%. or is it 3%? diveded by 6 months? I'm really confused by this one.

jim
It would be 4000 * 1.06^6, which is 5754.08 after six months.

Cathy
Thanks. And for b. I subtract 4000 from
5754.08 to get the interest? 
jim
Think of it this way:
After 1 month, you're 6% up, which is an extra 240.
Next month, you get 6% of 4240, which is an extra 254.40, and you throw that into the pot.
Next month, you start with 4494.4, and throw another 6% onto that again.
Working through an example like this month by month is boring, but the best way I know of to get to grips with the figures. 
jim
And for b. yes.

Cathy
Ok thank you. How did you get the 101.2%
for c. 
jim
(1.06^12  1) * 100

jim
Or you could so it this way:
Start with one dollar.
Add 6% to it.
Add 6% to that.
Add 6% to that.
and so on for 12 months.
At the end, you have $2.01 (ignoring rounding). $1 of that was your principal, so $1.01 is interest paid during the year, which is 101%. 
Cathy
Alright, I no I'm a little slow with this,
but can you explain the numbers
1.06^12  1)*100. 1.06 is the 6% right? what is the ^121 and 100. Sorry to be a pain, this is the first time I'm doing this type of thing. 
jim
Look at my second explanation.
That takes it one step at a time.
Each month, you have 1.06 as much as you had the month before, so at the end of the 12 months you have
1.06*1.06*1.06*1.06*1.06*1.06*1.06*1.06*1.06*1.06*1.06*1.06
as much as at the start. That is
1.06 to the power of 12 = 1.06^12
But you have to subtrac your original principal, which is not part of the interest.
As I say, the best way to see it is with a calculator. Enter your principal, and keep multiplying by 1.06 for each compounding period. 
Cathy
I'll give it a shot. Thanks so much for your help, I really appreciate it.

Cathy
Jim, quick question. What it I wanted to to invest the $4,000 in the highinterest savings account. Which is the savings account that pays 3% compounded daily.
How do you multiply the daily number? multiply the 4,000*3%*365 
jim
You have to take the
4000, and add 3% for the first day
and then add 3% to your answer for the second day
and then add 3% to your answer for the third day
How do you add 3%? Multiply by 1.03.
So just take your 4000 and multiply it by 1.03, and repeat the multiplication 365 times.
That is NOT the same as 4,000*3%*365.
What you want is 4000*1.03*1.03*1.03*1.03*1.03*1.03*1.03*1.03...
and so on. 
Cathy
Thanks again, really appreciate it.

Cathy
Wait, is there a way to do this on a calculator? Like 1.03*365

jim
Right. It's not 1.03*365.
It's 1.03 to the power of 365
aka 1.03^365
also written 1.03 with 365 written in superscript above and just to the right of 1.03.
The power key isn't on all calculators.
Scientific calculators usually have this function, and it's usually on a key marked x^y (like the Windows calculator in scientific mode) or a key with a large x and a smaller y above and to the right of it.
The Windows calculator (use View / Scientific) gives me
48482.724527501361858081469898097
as the answer to 1.03^365, if you want to check your numbers. 
jim
P.S. Yes, this would mean that if you started with $4,000, at the end of the year, you would have
$4,000 * 48482.72
= $193,930,880
(and you took the measly 6% per month? :) 
Cathy
Thanks so much Jim.

Cathy
What would I receive at the end of 6 months?

Cathy
What value would I receive at the end of 6 months?

jim
In case you ever come back to this, I just saw MathMate answering the same question, but reading "6% compounded monthly" as "6% _per annum_ compounded monthly".
This makes it a whole different question, and incidentally brings the interest rates back into the land of the normal.
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