Question

Kristina decided she would do better if she put her money in the bank for one year, which paid 3.25% interest compounded annually. The interest is calculated termly. Was she right that this invest will give her more money after 5 years?

Answer 1

To determine whether Kristina's decision to invest her money in the bank for one year at an interest rate of 3.25% compounded annually was the right choice, we need to compare the final amount after 5 years with the amount she would have earned if she didn't invest.

Given:

Interest rate (r) = 3.25% per year (0.0325 as a decimal)

Time (t) = 5 years

To calculate the final amount after 5 years with annual compounding, we can use the formula:

A = P * (1 + r)^t

Where:

A = Final amount

P = Initial principal (the amount invested)

Since we don't know the initial principal, we'll consider it as 1 for comparison purposes. Therefore, the formula becomes:

A = (1 + r)^t

Calculating the final amount after 5 years without investing:

A_without_investment = (1 + 0.0325)^5

Using a calculator or a computer, we can evaluate (1 + 0.0325)^5 to be approximately 1.17098.

Calculating the final amount after 5 years with the bank investment:

A_with_investment = (1 + 0.0325)^5

Again, using a calculator or a computer, we find that (1 + 0.0325)^5 is approximately 1.17098.

Since both calculations result in the same final amount, it means that whether Kristina invested her money or not, the end result after 5 years would be the same.

Therefore, Kristina's decision to invest her money in the bank for one year at a 3.25% interest rate compounded annually does not result in more money after 5 years compared to not investing.

Step-by-step explanation:

Answer 2

To determine whether Kristina's decision to invest her money in the bank with a 3.25% interest rate compounded annually is better than continuous compounding over 5 years, we need to calculate the future value under both scenarios.

First, let's calculate the future value with annual compounding.

The formula to calculate the future value (A) with annual compounding is:

A = P * (1 + r/n)^(n*t)

Where:

A = the future value including interest

P = the principal amount (initial deposit)

r = the interest rate (in decimal form)

n = the number of compounding periods per year

t = the number of years

In this case, Kristina's investment has an interest rate (r) of 3.25% (or 0.0325 in decimal form), and since the interest is calculated termly, there are 4 compounding periods per year (n = 4). The number of years (t) is 5.

Calculating the future value with annual compounding:

A = P * (1 + r/n)^(n*t)

= P * (1 + 0.0325/4)^(4 * 5)

= P * (1 + 0.008125)^20

Now, let's calculate the future value with continuous compounding, which we can use the formula mentioned in the previous answer:

A = P * e^(rt)

Calculating the future value with continuous compounding:

A = P * e^(rt)

= P * e^(0.0325 * 5)

To compare the two scenarios, we need to calculate the future value for both of them. Let's assume the principal amount (P) is $1 for simplicity.

Future value with annual compounding:

A = P * (1 + 0.008125)^20

Future value with continuous compounding:

A = P * e^(0.0325 * 5)

Calculating the future value with annual compounding:

A = 1 * (1 + 0.008125)^20

≈ 1.166363

Calculating the future value with continuous compounding:

A = 1 * e^(0.0325 * 5)

≈ 1.167972

After rounding to two decimal places, the future value of Kristina's investment with continuous compounding is approximately $1.17, while with annual compounding it's approximately $1.17 as well.

Therefore, both scenarios yield almost the same amount of money after 5 years, with a slight advantage to continuous compounding. Hence, Kristina was not entirely correct.