John loaned $72000 from the bank to be paid over 10 years. Given that the bank charges compound interest, calculate the annual interest rate if his monthly instalments over the 10 years was $890
Hello, Punch!
I believe this problem is unsolvable by ordinary means.
John loaned $72,000 from the bank to be paid over 10 years.
The bank charges compound interest,
Calculate the annual interest rate if his monthly installments was $890.
This is an Amortization problem.
$\displaystyle \text{Formula: }\;A \;=\;P\,\dfrac{i(1+i)^n}{(1+i)^n-1}$
. . $\displaystyle \text{where: }\;\begin{Bmatrix}A &=& \text{periodic payment} \\ P &=& \text{principal} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}$
We are given: .$\displaystyle A = \$890,\;P = \$72,\!000,\;n = 120\text{ months}$
We want the annual interest rate, $\displaystyle r.$
. . The periodic interest rate is: .$\displaystyle i = \frac{r}{12}$
Substitute: . $\displaystyle 890 \;=\;72,\!000\,\dfrac{i(1+1)^{120}}{(1+i)^{120}-1} $
Now all you have to do is solve for $\displaystyle i$ . . . then multiply by 12.
Good luck!
I'll wait in the car . . .
Easier to use P = Ai / (1 - f) where f = 1 / (1 + i)^n
Rearrange to get P/A = i / (1 - f)
"Hit and miss" (iteration) required to get i.
If i / (1 - f) > P/A then i is too high...
Keep adjusting i until both sides are reasonably equal...
Get my drift?
For this one, i = .702492.... is VERY close, hence .702492 * 1200 = ~8.43%
The only formula I have learnt about is $\displaystyle A=P(1+\frac{i}{100})^n$
$\displaystyle A= 890 *12 * 10$
$\displaystyle = 106800$
therefore, $\displaystyle A=P(1+\frac{i}{100})^n$
$\displaystyle 106800=72000(1+\frac{i}{100})^{10}$
$\displaystyle i = 4.02%$
why is my answer different from all of yours?
That's the formula to calculate the Future value of an amount invested now:
NOT of a series of deposits or payments.
Like, if $3000 is deposited NOW in an account paying 9% annually,
after 8 years the $3000 will have accumulated to:
3000(1 + 9/100)^8 = 5977.68792...$5977.69 rounded
That formula does NOT apply in your problem's case.
You were given the formula previously by Soroban and myself.
If you are unable to realise that the rate i CANNOT be solved for directly,
then you need classroom help which cannot be provided here ...