Question 1041884
your formula is p = a * e^(-rt)


p is the future value
a is the present amount
e is the scientific constant e
r is the annual growth rate as a decimal
t is the number of years


3.8% is the annual growth rate as a percent.
divide it by 100 to get .038
that's the annual growth rate as a decimal.


p = .76
a = 1


formula becomes .76 = 1 * e^(-.038 * t)


simplify this to get .76 = e^(-.038 * t)


take the natural log of both sides of the equation to get:


ln(.76) = ln(e^(-.038 * t))


since ln(e^(-.038 * t) is equivalent to -.038 * t * ln(e), your equation becomes:


ln(.76) = -.038 * t * ln(e)


since ln(e) is equal to 1, your equation becomes:


ln(.76) = -.038 * t


divide both sides of this equation by -.038 to get:


ln(.76) / -.038 = t


solve for t to get t = ln(.76) / -.038 = 7.222022255


that's your answer.


it will take 7.222022255 years for the purchasing power of 1 dollar to become .76 of a dollar if the inflation rate is 3.8% per year.


you can confirm the answer is correct by replacing t in the original equation with the answer and then evaluating the equation.


you will get:


.76 = 1 * e^(-.038 * 7.222022255)


after evaluation, you will get .76 = .76.


this confirms the solution is correct.