Question

HELP PLEASE? <3 You inherit $25000 and deposit it into an account that earns 4.5% annual interest compounded quarterly. How much money will be in the account after ten years?

Answer 1

Use the formula


\[\Large A = P\left(1+\frac{r}{n}\right)^{n*t}\]

Answer 2

In this case,

P = 25000
r = 0.045 (the decimal form of 4.5%)
n = 4 (compounding quarterly)
t = 10

Answer 3

I got 25000 (1+(0.045/4)^4(10)

Answer 4

now evaluate

Answer 5

When I calculate it comes out to 261441.2716. I'm not sure that seems right.

Answer 6

that's way too big

Answer 7

10 is in the exponent

Answer 8

so it should be 25000*(1+0.045/4)^(4*10)

Answer 9

evaluate that

Answer 10

Okay, so I put it in wrong. So, it is 39109.42163, right?

Answer 11

Very good,


\[\Large A = P\left(1+\frac{r}{n}\right)^{n*t}\]

\[\Large A = 25000\left(1+\frac{0.045}{4}\right)^{4*10}\]

\[\Large A \approx 39,109.4216347769\]

\[\Large A \approx 39,109.42\]

Answer 12

So you'll have $39,109.42 in the account in 10 years

Answer 13

Lovely~! Thank you. Would you mind if I ask you for help on one more part of the question set?

Answer 14

Earning 39,109.42 - 25,000 = 14,109.42 in interest

Answer 15

sure go ahead

Answer 16

If the interest were to be compounded continuously at 4.5%, how much money would be in the account after 10 years? And how many years you would reach your goal of 65,000, if the interest is compounded continuously.
I got 21.4 years for how long it takes to get 65000.

Answer 17

And thank you for helping me with all of this. <3

Answer 18

So you were able to answer the second part of this? but not the first part?

Answer 19

I did it on my calculator quickly after you helped me just now.

Answer 20

how did you do it on your calculator?

Answer 21

When you showed me how to put in the formula, I changed the year, putting in 20. Since 10 gives you around 39 grand, So I looked to see if 20 would double it, but when it didn't I looked to se how much more I would have to put in and came up with around 21.

Answer 22

Am I not supposed to do it like that? o.o

Answer 23

well notice how it says "compounded continuously"

so the formula is going to be (slightly) different

Answer 24

May I ask how so?

Answer 25

You will now use this formula

\[\Large A = Pe^{rt}\]

P is the amount you deposit or invest
r is the interest rate (in decimal form)
t is the time in years


'e' is a special constant (like pi)
e = 2.71828182846...

Answer 26

The explanation as to why you use this formula for compounding continuously is a bit lengthy and it involves some advanced theory

Here is the explanation

http://www-stat.wharton.upenn.edu/~waterman/Teaching/IntroMath99/Class04/Notes/node13.htm

Unfortunately it requires some background in calculus to fully understand how they're deriving the formula

Answer 27

anyways, you're taking the interest you've earned ($14,109.42) and you are depositing it into another account which continuously compounds interest for 10 years

So

P = 14,109.42
r = 0.045
t = 10

and you plug all that into

\[\Large A = Pe^{rt}\]

Answer 28

So, 14,109.42e^(0.045*10)?

Answer 29

Doing it like that I get 39765.7485.

Answer 30

what did you type in to get that?

that's too high

Answer 31

14,109.42x^(0.045*10). I can't find e on the calculator.

Answer 32

Would the x change it that much?

Answer 33

what kind of calculator do you have?

Answer 34

A graphing calculator. I could use the alpha, but I'm not sure how much of a difference that would make to change it to a e.

Answer 35

a texas instruments calculator?

Answer 36

Changing it to a e in the equation made it 0.
And yes. A TI-84 Plus.

Answer 37

see the "LN" button (under the "Log" button)?

Answer 38

Yes.

Answer 39

above the LN is "e^x"

so you hit the "2nd" key, then hit LN to get to e^x

Answer 40

Okay~! So, it would be 22127.97532.

Answer 41

good, that rounds to $22,127.98

Answer 42

Ahhh...~! Okay! Thank you so much! ^^

Answer 43

For the second part, the answer you got (21.4 years) is incorrect

Answer 44

Is it?

Answer 45

Yes, you need to solve for t in...


A = P*e^(r*t)

65000 = 14109.42*e^(0.045*t)

Answer 46

I got 0. o.o

Answer 47

hmm odd

Answer 48

how are you getting 0?

Answer 49

Not sure. I'm putting it in the way you wrote it down. Could it be the letter?

Answer 50

I'll show you how I solved for t.


A = P*e^(r*t)

65000 = 14109.42*e^(0.045*t)

65000/14109.42 = e^(0.045*t)

4.60685 = e^(0.045*t)

ln(4.60685) = ln(e^(0.045*t))

ln(4.60685) = 0.045*t*ln(e)

ln(4.60685) = 0.045*t*1

ln(4.60685) = 0.045*t

ln(4.60685)/0.045 = t

t = ln(4.60685)/0.045

t = 33.945429

Answer 51

the "ln" is really "LN" just with lowercase letters

Answer 52

So it takes roughly 33.945429 years to go from $14,109.42 to $65,000

Answer 53

That is so much. How do you retain it all? ;w;

Answer 54

tons and tons of practice

Answer 55

Oh, goodness. ><

Answer 56

The first step after you plug everything in is to isolate e^(0.045*t)

Answer 57

once you do that, you need to isolate t

Answer 58

and you use natural logs (LN) to do so

Answer 59

You explain this better than my teacher. o3o

Answer 60

Thanks. I'm glad it's clicking now.

Answer 61

Also, you'll use these log rules

http://www.purplemath.com/modules/logrules.htm

in this specific case, I used rule # 3

Answer 62

Okay. Thank you. :)

Answer 63

you're welcome

Answer 64

Can I ask you a general question? ><;

Answer 65

Just one more.

Answer 66

Go ahead

Answer 67

I am working on the next question right now and it asked me to identify the annual growth rate and the growth factor. What is the difference?

Answer 68

What's the full problem you're working on?

Answer 69

you deposit $2000 in an account that earns 5% annual interest compounded monthly.

Answer 70

There are two parts before that and I am on the last part.

Answer 71

how long are you depositing it for?

Answer 72

The first question was; What will your balance be after 2 years?
The second question; Estimate how long it would take for your investment to double.
I'm not sure if that tells you. It only says annually in the third question; Identify the annual growth rate and the growth factor.

Answer 73

it seems like this is a trick question because I think the annual growth rate is simply 5%

Answer 74

How would you come across that if you don't mind me asking?

Answer 75

it's given to you

the 5% APR

Answer 76

OH! Okay. I read over that. ^^;;

Answer 77

as for the annual growth factor, that would be

1 + r/n = 1 + 0.05/12 = 1.0041667

which is approximate

Answer 78

Okay~! That makes more sense. Thank you. <3

Answer 79

you're welcome