How long will it take for prices in the economy to double at a 6% annual inflation rate? Round to the nearest hundredth. Please explain how to solve this problem, thanks.
answered below, at length
To solve this problem, we need to use the formula for compound interest, which is used to calculate the future value of an investment or the increase in prices over time. The formula is:
FV = PV * (1 + r)^n
Where:
FV = future value
PV = present value
r = interest rate per period (in this case, the inflation rate)
n = number of periods
In this case, we want to find out how long it takes for prices to double, so the future value (FV) will be 2 times the present value (PV). The inflation rate (r) is given as 6% or 0.06, and we need to find the number of periods (n).
Substituting the values into the formula, we get:
2 = 1 * (1 + 0.06)^n
Next, we isolate the variable by dividing both sides of the equation by 1:
2/1 = (1 + 0.06)^n
Simplifying the left side of the equation:
2 = 1.06^n
To find the value of n, we need to take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):
ln(2) = ln(1.06^n)
Applying the power rule of logarithms:
ln(2) = n * ln(1.06)
Finally, we solve for n by dividing both sides of the equation by ln(1.06):
n = ln(2) / ln(1.06)
Using a calculator or software to evaluate this expression, we find that n is approximately 11.9.
Therefore, it will take approximately 11.9 years for prices in the economy to double at a 6% annual inflation rate.