Question

Taylor invested $280 in an account paying an interest rate of 8(3)/(4)% compounded continuously. Isaiah invested $280 in an account paying an interest rate of 8(1)/(2)% compounded quarterly. After 20 years, how much more money would Taylor have in her account than Isaiah, to the nearest dollar?

Answer 1

To find the difference in the amount of money Taylor and Isaiah would have after 20 years, we need to calculate the compound interest for each of them. Taylor would have about $160 more in her account than Isaiah.

Step-by-step explanation:

To find the amount of money Taylor and Isaiah would have after 20 years, we can use the formula for compound interest:

A = P*e^(rt)

Where A is the final amount, P is the initial principal, r is the interest rate (in decimal form), and t is the time in years.

For Taylor: P = $280, r = 8(3)/(4)% = 8.75% = 0.0875, t = 20 years. Plugging in these values, we have:

A = $280 * e^(0.0875*20) = $280 * e^(1.75) ≈ $1175.51

For Isaiah: P = $280, r = 8(1)/(2)% = 8.5% = 0.085, t = 20 years. Plugging in these values, we have:

A = $280 * (1 + 0.085/4)^(4*20) ≈ $1015.92

To find the difference in their accounts, we subtract Isaiah's amount from Taylor's amount:

Difference = $1175.51 - $1015.92 ≈ $159.59

To the nearest dollar, Taylor would have about $160 more in her account than Isaiah after 20 years.

Answer 2

Taylor would have approximately $43.10 more in her account than Isaiah after 20 years.

How to solve

Calculating Taylor's and Isaiah's Account Value:

Taylor's Account:

Interest rate: 8(3)/(4)% = 21/4% compounded continuously

Formula for continuous compounding: A = P * e^(rt)

Calculating Taylor's balance: A = $280 * e^((21/4 * 20)/100) ≈ $739.85

Isaiah's Account:

Interest rate: 8(1)/(2)% = 4% compounded quarterly

Number of compounding periods: 20 years * 4 quarters/year = 80 quarters

Formula for quarterly compounding: A = P * (1 + r/n)^n

Calculating Isaiah's balance: A = $280 * (1 + 4/80)^80 ≈ $696.75

Difference in Account Value:

Taylor - Isaiah = $739.85 - $696.75 ≈ $43.10

Therefore, Taylor would have approximately $43.10 more in her account than Isaiah after 20 years.