In a suburb, housing costs have been increasing at 5.2% per year compounded annually for the past 8 years. A house worth $260,000 now would have had what value 8 years ago?
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To find the value of the house 8 years ago, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value (current value of the house)
P = the present value (unknown)
r = the annual interest rate (5.2% or 0.052)
n = the number of times interest is compounded per year (compounded annually, so n = 1)
t = the number of years (8)
We know that the current value of the house is $260,000 (A).
260,000 = P(1 + 0.052/1)^(1*8)
Now, let's solve for P:
260,000 = P(1.052)^8
Divide both sides by (1.052)^8:
P = 260,000 / (1.052)^8
Calculating this expression, the value of the house 8 years ago would have been approximately $194,925.
To find the value of the house 8 years ago, we can use the concept of compound interest. The formula for calculating compound interest is:
Final Amount = Principal x (1 + Rate)^Time
Here, the Principal is the initial value of the house 8 years ago, the Rate is the annual increase in housing costs (5.2% or 0.052), and the Time is 8 years.
We know the current value of the house is $260,000.
So, we can rearrange the formula to solve for the Principal:
Principal = Final Amount / (1 + Rate)^Time
Plugging in the given values:
Principal = $260,000 / (1 + 0.052)^8
Calculating this using a calculator or spreadsheet software, the value of the house 8 years ago would be approximately $184,592.76.