# 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 6-month 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?

1. 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%

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.

3. jim

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

4. Cathy

Thanks. And for b. I subtract 4000 from
5754.08 to get the interest?

5. 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.

6. jim

And for b. yes.

7. Cathy

Ok thank you. How did you get the 101.2%
for c.

8. jim

(1.06^12 - 1) * 100

9. jim

Or you could so it this way:

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%.

10. 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 ^12-1 and 100. Sorry to be a pain, this is the first time I'm doing this type of thing.

11. 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.

12. Cathy

I'll give it a shot. Thanks so much for your help, I really appreciate it.

13. Cathy

Jim, quick question. What it I wanted to to invest the \$4,000 in the high-interest 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

14. jim

You have to take the

4000, and add 3% for the first 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.

15. Cathy

Thanks again, really appreciate it.

16. Cathy

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

17. 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.

18. 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? :-)

19. Cathy

Thanks so much Jim.

20. Cathy

What would I receive at the end of 6 months?

21. Cathy

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

22. 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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