Question

Fabian invested $6,800 in an account paying an interest rate of 2 1/8​ % compounded annually. Caleb invested $6,800 in an account paying an interest rate of 1 7/8 ​ % compounded continuously. To the nearest hundredth of a year, how much longer would it take for Caleb's money to triple than for Fabian's money to triple?

Answer 1

To find out how much longer it would take for Caleb's money to triple than Fabian's, we need to calculate the time it takes for each investment to triple. By using the compound interest formulas for each investment, we find that it would take approximately 17.41 years for Fabian's money to triple and 19.71 years for Caleb's money to triple. Therefore, it would take approximately 2.3 years longer for Caleb's money to triple than for Fabian's money to triple.

Step-by-step explanation:

To find out how much longer it would take for Caleb's money to triple than Fabian's, we need to calculate the time it takes for each investment to triple. Let's start with Fabian. We can use the formula for compound interest:

P = P0(1 + r/n)nt

where:

P is the final amount (triple the initial amount)

P0 is the initial amount

r is the interest rate (2 1/8% as a decimal)

n is the number of times compounded annually (1)

t is the time in years

By substituting the given values into the formula, we get:

3(P0) = (P0)(1 + 0.02125)1t

Simplifying the equation, we have:

3 = (1.02125)t

Taking the logarithm of both sides, we find:

t ≈ log1.021253

Using a calculator, we obtain:

t ≈ 17.41 years

Now let's find the time it takes for Caleb's money to triple. Since the interest is compounded continuously, we can use the formula:

P = P0ert

where:

P is the final amount (triple the initial amount)

P0 is the initial amount

e is Euler's number (approximately 2.71828)

r is the interest rate (1 7/8% as a decimal)

t is the time in years

By substituting the given values into the formula, we get:

3(P0) = (P0)e0.01875t

Simplifying the equation, we have:

3 = e0.01875t

Taking the logarithm of both sides, we find:

t ≈ loge3

Using a calculator, we obtain:

t ≈ 19.71 years

Therefore, it would take approximately 2.3 years longer for Caleb's money to triple than for Fabian's money to triple.

Answer 2

It would take Caleb's money about 0.98 years longer to triple compared to Fabian's money.

Certainly, let's break down the calculations step by step.

Fabian's Investment (compounded annually):

The formula for compound interest is A = P * (1 + r/n)^(nt), where:

P_F (principal) = $6,800,

r_F (annual interest rate) = 2 1/8 % or 0.02125 (convert percentage to decimal),

n_F (number of times compounded per year) = 1 (compounded annually),

A_F (future value) = 3 times the principal.

We want to solve for t_F:

3 * P_F = P_F * (1 + 0.02125/1)^(1 * t_F)

3 * 6,800 = 6,800 * (1 + 0.02125)^t_F

3 = (1.02125)^t_F

t_F ≈ ln(3)/ln(1.02125)

t_F ≈ 16.47 years

Caleb's Investment (compounded continuously):

The formula for continuous compound interest is A = P * e^(rt), where:

P_C (principal) = $6,800,

r_C (continuous interest rate) = 1 7/8 % or 0.01875 (convert percentage to decimal),

A_C (future value) = 3 times the principal.

We want to solve for t_C:

3 * P_C = P_C * e^(0.01875 * t_C)

3 * 6,800 = 6,800 * e^(0.01875 * t_C)

3 = e^(0.01875 * t_C)

t_C ≈ ln(3)/0.01875

t_C ≈ 17.45 years

Difference in Time:

Difference = t_C - t_F

Difference ≈ 17.45 - 16.47

Difference ≈ 0.98 years