Raising Maine magazine's August 5,2009, Home GREEN Home" article about going green discusses ways to improve a home's energy efficiency while keeping conservation in mind. From an efficient floor plan, to a house plan maximizing sunlight, triple-pane glass windows, radiant floor heating, and the use of recycled materials, the goal is to reduce the amount of fossil fuels consumed. Jim and Sonja Tanner were inspired by the article and are beginning the planning process for updating their home to reduce its environmental footprint. They received an initial quote of $23,500. What payment must they make at the end of each quarter at 8 % compounded quarterly for five years to reach their goal of $23,500?
solve for P , the payment
P(1.02^20 - 1)/.02 = 23500
(I got P =967.18 )
To find the payment Jim and Sonja Tanner need to make at the end of each quarter for five years to reach their goal of $23,500, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value of the annuity
P = Payment (unknown)
r = Interest rate per quarter (8% / 4 = 2% or 0.02)
n = Number of quarters (5 years * 4 quarters per year = 20 quarters)
We need to rearrange the formula to solve for P:
P = FV * (r / ((1 + r)^n - 1))
Plugging in the given values:
P = $23,500 * (0.02 / ((1 + 0.02)^20 - 1))
Now we can calculate the payment amount:
P = $23,500 * (0.02 / ((1.02)^20 - 1))
P = $23,500 * (0.02 / (1.485947 - 1))
P = $23,500 * (0.02 / 0.485947)
P = $23,500 * 0.041132
P ≈ $967.33
Therefore, Jim and Sonja Tanner must make a payment of approximately $967.33 at the end of each quarter for five years to reach their goal of $23,500.