David invested $340 in an account paying an interest rate of 2 1/8% compounded continuously. Natalie invested $340 in an account paying an interest rate of 2 3/4% compounded quarterly. After 17 years, how much more money would Natalie have in her account than David, to the nearest dollar?
To calculate the amount of money in each account after 17 years, we'll use the compound interest formula:
A = P * e^(rt)
Where:
A is the final amount of money
P is the principal amount (initial investment)
e is Euler's number (approximately 2.71828)
r is the interest rate (in decimal form)
t is the time in years
Let's start with David's account:
P = $340
r = 2 1/8% = 2.125% = 0.02125 (decimal form)
t = 17 years
Using the formula, we can calculate David's final amount (A_david):
A_david = 340 * e^(0.02125 * 17)
Now let's proceed with Natalie's account:
P = $340
r = 2 3/4% = 2.75% = 0.0275 (decimal form)
t = 17 years
Using the formula, we can calculate Natalie's final amount (A_natalie):
A_natalie = 340 * (1 + 0.0275/4)^(4 * 17)
Finally, to find the difference between the two accounts, we subtract David's final amount from Natalie's final amount:
Difference = A_natalie - A_david
Let's calculate it step by step:
First, David's final amount (A_david):
A_david = 340 * e^(0.02125 * 17)
Using a calculator, we find that A_david ≈ $631.20 (rounded to the nearest cent).
Now, Natalie's final amount (A_natalie):
A_natalie = 340 * (1 + 0.0275/4)^(4 * 17)
Again, using a calculator, we find that A_natalie ≈ $659.73 (rounded to the nearest cent).
Finally, the difference between Natalie's and David's accounts:
Difference = A_natalie - A_david
Difference ≈ 659.73 - 631.20 ≈ $28.53 (rounded to the nearest cent)
Therefore, to the nearest dollar, Natalie would have approximately $29 more in her account than David after 17 years.