Question

a woman paid $97,000 down for a house and agreed to pay 18 quarterly payments of $4,800 each. if money is worth 4% compounded quarterly, how much (in $) would the house have cost if she had paid cash? (round your answer to the nearest cent.)

Answer 1

The present value of the annuity, which is the total of 18 quarterly payments of $4,800 each at a 4% compounded quarterly interest rate, is approximately $76,192.32. Adding this to the down payment of $97,000, the house would have cost approximately $173,192.32 in cash.

Step-by-step explanation:

To find out how much the house would have cost if the woman had paid cash instead of making a down payment and subsequent quarterly payments, we need to calculate the present value of the payments using the formula for the present value of an ordinary annuity. Given that the interest rate is 4% compounded quarterly, we can find the present value of the 18 quarterly payments of $4,800 each. Once we have this present value, we can add it to the down payment of $97,000 to get the total cash price.

The present value (PV) of the annuity is given by:

PV = Pmt * ((1 - (1 + r)^-n) / r)

Where:

• Pmt = quarterly payment of $4,800
• r = quarterly interest rate (0.04 / 4 = 0.01)
• n = total number of payments (18)

Using this formula:

PV = $4,800 * ((1 - (1 + 0.01)^-18) / 0.01) = $4,800 * ((1 - (1.01)^-18) / 0.01)

After calculating the present value of the annuity, we add it to the initial down payment to find the total cash price of the house.

Let's do the math:

PV ≈ $4,800 * 15.8734 ≈ $76,192.32

Therefore, the total cash price = down payment + present value of the payments = $97,000 + $76,192.32 ≈ $173,192.32

So, if the woman had paid cash, the house would have cost approximately $173,192.32.