Hello and HELP!!! U Mathematical GURUS!...

[SittingBull]

My niece needs to know how to solve this problem for a test.
Your help would be deeply appreciated!

Problem
u have saved $7000.00 toward the purchase of a new car costing 9000.00.
How long will the 7000.00 have to be invested at 9% compounded monthly to grow to 9000.00?

Forumlas,explanation of variables,etc... ,please.
Thanks
HappyPokering, [img]/images/graemlins/smile.gif[/img]
SittingBull

[well]

This is not really hard, did you try it yourself?

$(t): amount of dollars after t months, $(0)=7000
g: growrate, (100%+9%)/100% = 1.09

Now $(t)=$(0)*g^t, the standard exponential growth formula.
We want to know when we can buy the car, or: $(t)=9000

9000 = 7000 * 1.09 ^ t
{ division by 7000 }
9/7 = 1.09 ^ t
{ e^log(x) = x }
9/7 = ( e ^ log(1.09) ) ^ t
{ ( x ^ y ) ^ z = x ^ zy }
9/7 = e ^ t*log(1.09)
{ log, log(e ^ x) = x }
log(9/7) = t*log(1.09)
{ division by log(1.09) }
log(9/7) / log(1.09) = t

Hence, t = 2.916...

The rent is compounded monthly, so after 3 months we can buy the car.
We even have about $65 left!

I hope now you can explain it to her.

Next Time

[BruceZ]

The 9% is most likely per year, not per month.

[SittingBull]

[BruceZ]

It can be compounded quarterly, monthly, daily, hourly, or instantaneously, and the interest rate can still be given as yearly. Interest rates are almost always quoted per year unless otherwise noted. Where are you going to get 9% monthly interest? It's probably 9% per year compounded monthly. Make sure.

[Angel]

Instantly? 'e'-gads.

cheap math joke...sorry [img]/images/graemlins/wink.gif[/img]

[SittingBull]

I want to thank U for your i/p into my great-niece's prob.
SittingBull

[SittingBull]

I appreciate your complete explanation of this proble.
I dont want to aggravate u about problems that are trivial TO U. However,I'm asking your permission to consult U on future problems of this nature.
I do understand if u prefer NOT to be consulted about these types of problems in the future. I WILL honor your request. And I will let my nephew know that u gave me these solutions. They were NOT my solutions.
SittingBull
PS I'll convey the solution to my niece. She is having a test Monday. Hence,it's sort of a cram job.
My nephew,my niece's uncle, consulted me today about the problems today.
Thanks again
SittingBull

[SittingBull]

U secured a 35-YR loan to buy a home costing 150K. U make a 15% down payment.. The bank gives U a 7.3% annual rate compounded monthly.
What's your monthly payment and your equity after 18 years?
Thanks,
SittingBull

[BruceZ]

This was a useful exercise. It was the first time I've tried to derive the mortgage equations. Your niece and nephew should be having this fun instead of me having fun doing their homework. [img]/images/graemlins/smile.gif[/img] They may have been given some different formulas or methods to use other than what I've done here, I don't know. I'm depending only on the geometric series.

I checked my results below with Excel which has built in functions for doing this kind of thing, and it gives exactly the same answers I derived.

You have a loan of $127,500 at 7.3% for 35 years. If you just paid (7.3%/12)*127,500 = $775.63 each month, that would only cover the interest. If we pay more than this each month by an amount p, then the balance owed after the first payment will decrease by p, and after the second payment it will decrease by p + the interest we save on p, or p + (7.3%/12)*p, etc. In 35 years there will be 35*12 = 420 payments, and the total balance paid down will be:

sum{n = 0 to 419} p*(1 + 0.073/12)^n = 127,500

From the formula for the geometric series, this is:

p*[1 - (1 + 0.073/12)^420] / [1 - (1 + 0.073/12) ] = 127,500

p = $65.88694

So our total monthly payment is: $775.63 + $65.89 = $841.51/mo.

Excel gives exactly the same answer by using the function =PMT(7.3%/12,35*12,127500).

After 18 years, our equity will be, from the last equation evaluated at 18*12 = 216 payments:

equity = 65.88694*[1 - (1 + 0.073/12)^216 / (-.073/12) ] = $29,310.62.

Excel gives exactly the same answer by using the function =CUMPRINC(7.3%/12,35*12,127500,1,18*12,0). I had to load a special analysis tool pack to get this function as it explains in the help. It took me longer to find this function than to derive it.

Generalizing the above, we have derived this formula for the monthly payment:

monthly payment = P*i/n*{1 + 1/[ (1 + i/n)^(nT) - 1 ]}

where P = principal amount borrowed
i = interest rate per year
n = number of times compounded per year
T = length of the loan in years


The second term is p = (P*i/n) / [ (1 + i/n)^(nt) - 1 ]. If we substitute this into the expression for equity above, we get:

equity = P*[ (1 + i/n)^(nt) - 1 ] / [ (1 + i/n)^(nT) - 1 ]

Where t is the time we have been making payments in years,
and T is the total length of the loan. Again, P is the amount of the loan, and n is the number of times it is compounded per year.