Respuesta :
The main formula used in this problem is the Present
Value of an annuity Formula:
[tex]\displaystyle{ P=R[\frac{1-(1+i)^{-n}}{i}][/tex]
where:
The maximal monthly payment that Bethany and Samuel can afford is $1,500.
The best interest rate is %5.1 annually, and they want to sign a 30-year mortgage.
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thus
R=1,500
That is,
i) they will make 360 monthly payments,
ii) the largest amount of money they can borrow can be found using the formula:
[tex]\displaystyle{ P=R[\frac{1-(1+i)^{-n}}{i}][/tex]
substituting the known values:
[tex]\displaystyle{ P=1,500\cdot[\frac{1-(1+0.00425)^{-360}}{0.00425}]=276,268.65[/tex] dollars
the calculation can be performed using powerful calculating engines like wolframalpha or a scientific calculator.
iii) Since Bethany and Samuel also have $25,000 for a down payment, they can buy a house with a maximum cost of:
276,268.65+25,000=301,268.65 $, which we can round to 301,300$ for the sake of common sense.
iv)
If the house they want to buy has a price of $280,000, they can afford it.
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For the second part of the problem, we use the
$280,000-$25,000=$255,000
the monthly interest rate is the same, that is:
[tex]\displaystyle{ P=R[\frac{1-(1+i)^{-n}}{i}][/tex]
where:
P: present value (the mortgage value)
R: periodic payment
t: time (in years)
r: annual interest rate
m: number of compoundings per year
i=r/m (rate per period)
n=mt the total number of payments
The maximal monthly payment that Bethany and Samuel can afford is $1,500.
The best interest rate is %5.1 annually, and they want to sign a 30-year mortgage.
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thus
R=1,500
t=30
r=%5.1 = 5.1 /100 = 0.051
m=12, because they make the payments monthly
i=r/m=0.051/12=0.00425
n=mt = 12*30=360
------------------------------------------------------------------------------------------------That is,
i) they will make 360 monthly payments,
ii) the largest amount of money they can borrow can be found using the formula:
[tex]\displaystyle{ P=R[\frac{1-(1+i)^{-n}}{i}][/tex]
substituting the known values:
[tex]\displaystyle{ P=1,500\cdot[\frac{1-(1+0.00425)^{-360}}{0.00425}]=276,268.65[/tex] dollars
the calculation can be performed using powerful calculating engines like wolframalpha or a scientific calculator.
iii) Since Bethany and Samuel also have $25,000 for a down payment, they can buy a house with a maximum cost of:
276,268.65+25,000=301,268.65 $, which we can round to 301,300$ for the sake of common sense.
iv)
If the house they want to buy has a price of $280,000, they can afford it.
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For the second part of the problem, we use the
Compound interest formula:
[tex]A=P(1+ \frac{r}{m} )^{mt}=P(1+ i)^{n} [/tex]
where
A= Future value
P =
the Principal (the initial amount of money)
If the couple buys a house of value $280,000 and they own
$25,000 (the down payment) then the mortgage is :$280,000-$25,000=$255,000
the monthly interest rate is the same, that is:
i=0.00425
n=360
Applying the formula we have:
[tex]A=P(1+ i)^{n} \\\\A=255,000\cdot(1+ 0.00425)^{360}=1,173,820[/tex]
this is the total value that has to be payed in 360 monthly payments,
thus every month the couple will pay $1,173,820/360= $3,260.61
after the firs 12 payments, t=1, the future value becomes:
[tex]A=P(1+ \frac{r}{m} )^{mt}\\\\A=255,000(1+0.00425 )^{12}=268,313[/tex]
thus the value is 268,313, so the balance of the mortage is
268,313-12*3,260.61=229,185.68 dollars
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Last part: YES
"the values of i, P, and n for a four-year loan of $12,000 at an annual interest rate of 4.2%."
are
P: $12,000 (principal)
t: 4
r: 42/100=0.042
m: 12
i=0.042/12= 0.0035
n=mt=12*4=48
