Logarithms Calc

Question

Using the formula PV= R[1-(1+i)^-n where the right side of the formula is being divided by i. Suppose a bank offers you a 3.4% interest rate, compounded semi-annually, on a mortgage amount of $173112. You pay $1000/mo, or $6000 semi annually. How many semi-annual payments would you have to make to pay off the mortgage. How long, in years, would it take to pay off the mortgage?

Answer 1

First, we need to rearrange the formula to solve for n:

PV = R[1 - (1 + i)^(-n)] / i

Multiplying both sides by i and dividing by R, we get:

iPV / R = 1 - (1 + i)^(-n)

Subtracting iPV / R from both sides, we get:

(1 + i)^(-n) = 1 - iPV / R

Taking the reciprocal of both sides, we get:

(1 + i)^n = 1 / (1 - iPV / R)

Taking the natural logarithm of both sides, we get:

n ln(1 + i) = ln(1 / (1 - iPV / R))

Dividing both sides by ln(1 + i), we get:

n = ln(1 / (1 - iPV / R)) / ln(1 + i)

Now we can plug in the given values:

i = 0.034 / 2 = 0.017 (since interest is compounded semi-annually) PV = $173112 R = $6000 R / 12 = $1000 (since you pay $1000/mo) n = number of semi-annual payments

Plugging in these values, we get:

n = ln(1 / (1 - 0.017 * 173112 / 6000)) / ln(1 + 0.017)

Simplifying the expression inside the natural logarithm, we get:

n = ln(1.1136) / ln(1.017)

Using a calculator, we get:

n ≈ 42.6

Therefore, you would have to make 43 semi-annual payments to pay off the mortgage.

To find the time in years, we can divide the number of payments by 2 (since each payment is made every 6 months):

Time = n / 2

Time = 21.3

Therefore, it would take approximately 21.3 years to pay off the mortgage.