You have $100 and could use it to buy 80 songs from itunes. Instead, you invest your $100 at an annual effective interest rate of 11.3% for 4 years. The price of songs on itunes is subject to an annual effective rate of inflation of r%. At the end of 4 years, you can buy 101 songs.
Determine r .
To determine the annual effective rate of inflation, we can use the concept of future value.
Let's start by calculating the future value of your $100 investment after 4 years at an annual effective interest rate of 11.3%. The future value (FV) can be calculated using the formula:
FV = PV * (1 + r)^n
Where:
PV = Present value (initial investment)
r = Annual effective rate of interest
n = Number of years
In this case, the present value (PV) is $100, the number of years (n) is 4, and the future value (FV) is the amount needed to buy 101 songs from iTunes.
So, we have the equation:
FV = 100 * (1 + 0.113)^4
Now, we need to find the annual effective rate of inflation (r). Since the price of songs on iTunes is subject to inflation, we can rearrange the formula to solve for r:
r = (FV / PV)^(1/n) - 1
Substituting the known values:
r = (101 / 100)^(1/4) - 1
Calculating:
r = 1.006301 - 1
r = 0.006301
Therefore, the annual effective rate of inflation (r) is approximately 0.6301%.