Compounded continuosly

Question

Following the birth of a​ child, a parent wants to make an initial investment P0

that will grow to $50,000 for the​ child's education at age 17. Interest is compounded continuously at 6​%. What should the initial investment​ be? Such an amount is called the present value of $50,000 due 17 years from now.

Answer 1

-When dealing with interest problems, there are three main formulas you need to know. In this problem, the key words are "compounded continuously," and so we need to use the following formula: A=Pe^rt (in plain English, this formula means -> the amount of money you have in an account equals the principal amount times "e" raised to the power of "rt").

-An explanation of each variable is provided below. It is first and foremost very important to know what each variable means. When you do any word problem of this type, you have to look for the numbers in the word problem that are related to each variable.

A = amount of money that you have in an account after the principal(starting amount) has been sitting and has grown in the account after a certain number of years.

P = the principal amount of money you start with in the account.

e = approximately 2.718 (it is a special mathematical number, and you have an "e" button on your calculator for this number)

r = rate of interest (%) -> always convert it to a decimal when using it in your calculations.

t = time in years -> sometimes you have to convert days/months/etc. into years, so be sure you pay attention to how time is indicated in the word problem.

-Now, let's get to the word problem. After reading through it, identify your variables and plug them into the formula.

"Following the birth of a​ child, a parent wants to make an initial investment P

that will grow to $50,000 for the​ child's education at age 17. Interest is compounded continuously at 6​%. What should the initial investment​ be? Such an amount is called the present value of $50,000 due 17 years from now."

You can see I italicized and underlined the key parts. What do they mean?

initial investment P -> the money you start with in the account, which we don't know, so we leave it as "P" in the equation

grow to $50,000 -> how much money we want after a certain amount of time, so this is your "A" in the equation

at age 17 -> this piece of information helps you find what the variable "t" will be in the equation. Well, the word problem says "following the birth of a child," so that means t is 17 because 17 years will pass after the birth of the child.

compounded continuously at 6​% -> ***remember***, convert to decimal -> 0.06 -> this is your "r" in the equation

Let's put it all together and solve for "P":

A = Pe^rt

50000 = Pe^(0.06*17)

50000 = Pe^(1.02)

50000 = P(2.773194764) *remember to use your "e" button on your calculator to get this decimal

18029.74701 = P

Finally an answer! What does it mean in relation to the word problem? You would have to put $18,029.75 into the child's account at birth in order for it to grow to $50,000 by the time they are 17 years old.

Or more simply -> The initial investment P = $18,029.75

You did it!