Question

Alang invested $47,000 in an account paying an interest rate of 4 1/2% compounded annually. Amelia invested $47,000 in an account paying an interest rate of 3 7/8% compounded continuously. After 18 years, how much more money would Alang have in his account than Amelia, to the nearest dollar?

Answer 1

After 18 years, Alang would have about $9388.00 more money in his account than Amelia.

Explanation:

Step 1: Find amount in Alang's account after 18 years:

The formula for compound interest is given by:

A = P(1 + r/n)^(nt), where

• A is the amount in the account,
• P is the principal (aka investment),
• r is the interest rate (always a decimal),
• n is the number of compounding period per year,
• and t is the time in years.

Step 2: Identify values for compounded interest formula.

We can start by identifying which values match the variables in the compound interest formula:

• We don't know the amount, A, and must solve for it,
• the principal is $47000,
• 4 1/2% as a decimal is 0.045,
• n is 1 as the money is compounded annually and thus it only happens once per year,
• and t is 18.

Step 3: Plug in values and solve for A, the amount in Alang's account after 18 years:

Now we can plug everything into the compound interest formula to solve for A, the amount in Alang's account after 18 years:

A = 47000(1 + 0.045/1)^(1 * 18)

A = 47000(1.045)^18

A = 103798.502

A = $103798.50

Thus, the amount in Alang's account after 18 years would be about $103798.50.

Step 4: Find amount in Amelia's account after 18 years:

The formula for continuous compound interest is given by:

A = Pe^(rt), where

• A is the amount in the account,
• e is Euler's number,
• r is the interest rate (always a decimal),
• and t is the time in years.

Step 5: Identify values for continuous compounded interest formula:

We can start by identifying which values match the variables in the continuous compound interest formula:

• We don't know the amount, A, and must solve for it,
• P is $47000
• 3 7/8% as a decimal 0.03875,
• and t is 18.

Step 6: Plug in values and solve for A, the amount in Amelia's account after 18 years:

A = 47000e^(0.03875 * 18)

A = 47000e^(0.6975)

A = 94110.05683

A = 94110.06

Thus, the amount in Ameila's account after 18 years would be about $94410.06.

STep 7: Find the difference between amounts in Alang and Ameila's account after 18 years:

Since Alang would have more money than Ameila in 18 years, we subtract her amount from his to determine how much more money he'd have in his account than her.

103798.50 - 94410.06

9388.44517

9388

Therefore, after 18 years, Alang would have $9388.00 more money in his account than Amelia.