Question

How much interest (to the nearest dollar) would be saved on the following loan if the condominium were financed for 15 rather than 30 years? A $256,000 condominium bought with a 30% down payment and the balance financed for 30 years at 3.05%

Answer 1

Answer:

The interest saved is $49569.228 or $49569.

Explanation:

Total price of Condominium=$256,000

Downpayment=30% of total price=30%x$256,000= 76800

Amount Financed=Total Payment-Downpayment

Amount Financed=256000-76800=179200

Annual Interest rate=3.05%

Monthly interest rate =$\frac{3.05\%}{12}$=0.25146%

The montly installment is calculated as follows:

$M=\dfrac{P}{\dfrac{1-\left(\dfrac{1}{1+\dfrac{r}{100}}\right)^{nt}}{\dfrac{r}{100}}}$

Here

• M is the montly installment
• P is the amount financed
• r is the montly rate in percentage
• n is the number of years
• t is the number of months in a year

Case 1 when the number of years is 30.

So the equation becomes

$M=\dfrac{P}{\dfrac{1-\left(\dfrac{1}{1+\dfrac{r}{100}}\right)^{nt}}{\dfrac{r}{100}}}\\\\M=\dfrac{179200}{\dfrac{1-\left(\dfrac{1}{1+\dfrac{0.25146}{100}}\right)^{30*12}}{\dfrac{0.25146}{100}}}\\\\M=\dfrac{179200}{\dfrac{1-\left(\dfrac{1}{1+0.0025146}\right)^{30*12}}{0.0025146}}\\\\M=\dfrac{179200}{\dfrac{1-\left(\dfrac{1}{1.0025146}\right)^{30*12}}{0.0025146}}\\\\M=\dfrac{179200\times {0.0025146}}{1-\left(\dfrac{1}{1.0025146}\right)^{30*12}}\\M=\dfrac{450.61632}{0.59510 }\\M=\ 757.2087$

So the total amount paid in installments is

$T=M\times n\times t$

So the equation becomes

$T=M\times n\times t\\T=757.2087\times 30\times 12\\T=\ 272595.132$

So the interest is given as

$I=T-P\\I=272595.132-179200\\I=\ 93395.132$

So a total interest of $93395.132 is paid when the amount is financed for 30 years.

Case 2 when the number of years is 15.

So the equation becomes

$M=\dfrac{P}{\dfrac{1-\left(\dfrac{1}{1+\dfrac{r}{100}}\right)^{nt}}{\dfrac{r}{100}}}\\\\M=\dfrac{179200}{\dfrac{1-\left(\dfrac{1}{1+\dfrac{0.25146}{100}}\right)^{15*12}}{\dfrac{0.25146}{100}}}\\\\M=\dfrac{179200}{\dfrac{1-\left(\dfrac{1}{1+0.0025146}\right)^{15*12}}{0.0025146}}\\\\M=\dfrac{179200}{\dfrac{1-\left(\dfrac{1}{1.0025146}\right)^{15*12}}{0.0025146}}\\\\M=\dfrac{179200\times {0.0025146}}{1-\left(\dfrac{1}{1.0025146}\right)^{15*12}}\\M=\dfrac{450.61632}{0.36368 }\\M=\ 1239.0328$

So the total amount paid in installments is

$T=M\times n\times t$

So the equation becomes

$T=M\times n\times t\\T=1239.0328\times 15\times 12\\T=\ 223025.904$

So the interest is given as

$I=T-P\\I=223025.904-179200\\I=\ 43825.904$

So a total interest of $43825.904 is paid when the amount is financed for 15 years.

The savings on interest if the condominium is financed for 15 years is given as

$S=I_{30}-I_{15}\\S=93395.132-43825.904\\S=49569.228$

The interest saved is $49569.228 or $49569.