Please Help! College Costs MATH Business Questions

[retamar]

Please, I'm stucked on this problem. I need help ASAP.

The cost of a 4-year college education at a public university is expected to be $7,000 a year in 18 years. How much money should be invested now at 7% so that the balance of the account after 18 years covers the cost of a college education? (assume that the payments for college are made once a year for four years, so that if, e.g., college costs $N per year in Y years, that a payment of $N will be made in Y years from now; another payment of $N will be made Y+1 years from now; etc)

The answer is $7,506.13, but I do not know how I can get it.

Help please!

Thank you

 

[mmm4444bot]

What is the compounding period for the interest earned on this account ?

What formulas have you learned, so far ?

There seems to be at least one contradictory statement, in this exercise. Perhaps, I misunderstand the meaning of symbols N and Y. I'm reading the scenario as:

The account is opened with a single deposit today, and you're asking how to determine that this deposit is $7,506.13

18 years from today, $7,000 will be withdrawn

19 years from today, $7,000 will be withdrawn

20 years from today, $7,000 will be withdrawn

The amount left in the account at this point will earn sufficient interest over the next year to leave a balance of $7,000 (maybe a little more) for withdrawal 21 years from today, for the final payment to the college.

Is this a correct read ?

Because this read contradicts the statement that the balance in the account 18 years from today be $28,000 (i.e., the cost of four years at $7,000 per year).

I'm also wondering why the symbols N and Y are used. It seems to me that N is the constant 7000 because that's what the first sentence says. Since $7,000 is needed each year for four years beginning 18 years from now (Y), why not say so ?

I'm easily confused. Please clarify these points, for me.

 

[Denis]

The cost of a 4-year college education at a public university is expected to be $7,000 a year in 18 years. How much money should be invested now at 7% so that the balance of the account after 18 years covers the cost of a college education? (assume that the payments for college are made once a year for four years, so that if, e.g., college costs $N per year in Y years, that a payment of $N will be made in Y years from now; another payment of $N will be made Y+1 years from now; etc)
The answer is $7,506.13, but I do not know how I can get it.

$7,506.13 is correct IF the $7000's are withdrawn at end of years 18,19,20,21.
So problem is wrongly stated; should be: "balance of the account after 17 years"

 

[retamar]

Yeah! This is the correct reading. However, I'm not getting the 7,506.13 needed, and my result is always around 7,900!

This problem is on the annuity chapter, so I have 4 formulas: P_N = P(((1+i)^n) - 1)/i, P_0 = P(1-((1+i)^-n))/i, P_0 = (P*(1+i))*(1-((1+i)^-n))/i (annuity due), P_N = (P*(1+i)*(((1+i)^n) - 1) (annuity due).

 

[Denis]

7000(1 - 1/1.07^4) / .07 = 23710.4787...

23710.4787... / 1.07^17 = 7506.1303...

 

[retamar]

Thank you very much!

 

[Messagehelp]

Question is correct and

answer provided is wrong..

correct answer is as follows..


Detailed Explanation:
First Analyze the Problem Properly and it will be easy to do that.
After 18 years he Earned X dollars.
So in 18th year he has to pay $7000 to Wards College and x-7000 left by that time and he earn interest on that.
At 19th year he has left with (x-7000) * 1.07 = k (let say)
So in 19th year he has to pay $7000 towards college and k-7000 left by that time and he earn interest on that .
At 20th year he has left with (k-7000) * 1.07 = z (let say)
So in 20th year he has to pay $7000 towards college and z-7000 left by that time and he earn interest on that.
In 21st year he has to pay $7000 towards college and this amount is final.
So in 21st year he has left with $7000 which is equal to (z-7000) * 1.07
Thus solving for z we get, z = (7000/1.07) + 7000 = $13542.05607.
Now solving for K = z/1.07 + 7000 = (13542.05607/1.07) + 7000 = $19656.12717.
Now Solving for x = k/1.07 + 7000 = (19656.12717 / 1.07) + 7000 = $ 25370.21231
SO Ultimately we need 25370 dollars at end of 18 years in order to pay college fees.
Now Question changed to like this.
How much money should be deposited paying a yearly interest rate of 7% compounded yearly so that after 18 years the accumulated amount will be $25370?
For this we have to use present formula for compound interest given by P = A (1+i) ^ -n
I = 0.07 and n = 18
A= 25370.21231 * 1.07 ^ -18 = $7506.130372. = P present value (this we have to invest.)
$7506.130372 this is the required answer.
And Coming to first part that is summation of geometric series like this
7000 + 7000/1.07 + 7000/1.07^2 + 7000/1.07^3 = $25370.21231 = 7000 * (1-1.07^-4) / (1-1.07^-1)
Sum of geometric progression formula is a* (1-r^n) / (1-r) r is common ratio and less than 1.
NOTE : ALL Problems are NOT Solved Simply BY USING Formulas …..

 

[Denis]

> SO Ultimately we need 25370 dollars at end of 18 years in order to pay college fees.

You are taking the long road, plus getting exactly (without realising it!) what I did;
I show: 7000(1 - 1/1.07^4) / .07 = 23710.4787...
That's 17th year: but 23710.4787... * 1.07 = ~25370 = 18th year (as you show).
AND the 1st $7000 payment is taken THEN, leaving ~18370

> NOTE : ALL Problems are NOT Solved Simply BY USING Formulas …

True. BUT in this case, only 2 formulas needed, and NOTHING ELSE :shock:

Here's EXACTLY what the "account" will look like:

Year    Payment   Interest      Balance
  0                             7,506.13
  1                 525.43      8,031.56
.....
 17               1,551.15     23,710.48 (amount I show)
 18   -7,000.00   1,659.73     18,370.21 (~25,370 before the payment)
 19   -7,000.00   1,285.91     12,656.12
 20   -7,000.00     885.93      6,542.05
 21   -7,000.00     457.95           .00

 

[Denis]

And Coming to first part that is summation of geometric series like this
7000 + 7000/1.07 + 7000/1.07^2 + 7000/1.07^3 = $25370.21231 = 7000 * (1-1.07^-4) / (1-1.07^-1)

That's correct BUT is the way assuming an ANNUITY IMMEDIATE:
that will only confuse the student that posted the original problem.

We really want:
7000/1.07 + 7000/1.07^2 + 7000/1.07^3 + 7000/1.07^4 = 23710.48
FORMULA: 7000[1 - 1.07^(-4)] / .07 = 23710.48
which is usually shown in standard form: P[1 - (1 + i)^(-n)] / i

Now we look at this as a deposit of $23710.48 to accomodate 4 payments
of $7000, the 1st being a year later:

  0                            23,710.48
  1   -7,000.00   1,659.73     18,370.21
  2   -7,000.00   1,285.91     12,656.12
  3   -7,000.00     885.93      6,542.05
  4   -7,000.00     457.95           .00

 

[retamar]

THANK YOU!