Question

Solve using the formula for the future value of an ordinary annuity.

Starting in January, you make monthly payments at the end of each month into an account paying the specified yearly interest rate. Interest is compounded monthly. How much will you have available at the specified date? Do not round until the final answer. Then, round to the nearest cent.

Monthly payment, $225; yearly interest rate, 10%; date, September 1.

Answer 1

Question is: How many payments are made, then?

Answer 2

If you start at January, and end at September the first, how many months will have ended by then?

Answer 3

So ten months?

Answer 4

How do you figure that?

Answer 5

January...February... ... ... ... September = how many months?

Answer 6

7?

Answer 7

wait
imconfused 1

Answer 8

I think you need to recite your months of the year :3

Answer 9

I think I need a new brain

Answer 10

Or a persistent study-buddy :)
now... January... February...what next? :3

Answer 11

terenzreignz keep in mind that the payments are made at the end of the month, so september doesn't count since you want to know the balance on september 1st

Answer 12

I was going to tell her that as soon as she tells me how many months to september, but thanks for the reminder, anyway, @jim_thompson5910 :)

Answer 13

yw just wanted to make sure

Answer 14

trust me @hannie ... counting months is the least of your problems :)

Answer 15

wait so I am counting jan to sept?
isnt that 9 am i counting September

Answer 16

Yes, 9... *counting september*
But as jim_thompson mentioned, September shouldn't be counted as we are counting the balance on September 1... meaning September has not ended yet, and we know that payments are made during the ends of the months.

So you make 8 payments all in all (end of January until end of August, which is virtually the same as the beginning of September, as far as annuities are concerned)

So...
so far so good?

Answer 17

yes

Answer 18

Is this the answer ..or am I completely wrong 2573.07

Answer 19

I don't know. But I guess we're about to find out :)

Answer 20

Now, we also have to know the effective rate of interest... PER MONTH. This is probably the trickiest step, so pay attention. We are given the effective rate of interest PER YEAR.

Let's denote the YEARLY interest with \(\large\color{red}r\) and we'll denote the MONTHLY interest with \(\large\color{blue}i\) We need the value of \(\large\color{blue}i\), so I suggest you make use of technology to find its value using this relation:
\[\Large \left(1+\frac{\color{blue}i}{12}\right)^{12}=1+\color{red}r\]

Answer 21

Not forgetting that in this scenario, \[\Large \color{red}r = 10\%=0.1\]

Answer 22

actually, we're more interested in the value of \[\Large \frac{\color{blue}i}{12}\]
Either way, find the value.

Answer 23

okay

Answer 24

Just ring me when you find the value. Don't take too long, though, I have to go in a bit.

Answer 25

\[\Large \left(1+\frac{\color{blue}i}{12}\right)^{12}=1.1\]

Answer 26

So I am solving for i

Answer 27

You could, but you could also solve for just \[\Large \frac{\color{blue}i}{12}\].
That's the key value here.
Correction, \(\color{blue}i\) is NOT the monthly interest, but rather, the annual interest which is COMPOUNDED monthly.
the actual effective rate of interest per month is \(\Large\frac{\color{blue}i}{12}\)

So, I suggest just leaving it at \(\Large \frac{\color{blue}i}{12}\)

Answer 28

Might as well remind you that solving for i (or i/12) is simply a means to an end. that's not the answer yet, though from there, it can be done in just one more step.

Answer 29

Once you've solved for \(\color{blue}i\) or \(\Large \frac{\color{blue}i}{12}\)
The answer would be given by this formula:
\[\Large P\cdot s_n\]
Where
\[\Large s_n = \frac{\left(1+\frac{\color{blue}i}{12}\right)^{n}-1}{\frac{\color{blue}{i}}{12}}\]

Where
P is the amount paid at the end of each month
i is the annual interest compounded monthly and
n is the number or payments made.