Question

Mary needs to borrow $1000. She can borrow the money at 5.5% simple interest for 3 yr or she con borrow at 3% with interest compounded continuously for 3 yr
(a) How much total interest would Mary pay at 5.5% simple interest?
(b) How much total interest would Mary pay at 3% interest compounded continuously?
(c) Which option results in less total interest?

Answer 1

To determine the total interest Mary would pay in each scenario, we can calculate as follows:

### (a) Total interest with 5.5% simple interest for 3 years

The formula for simple interest is:
\[
I = P \times r \times t
\]
where:
- \( I \) is the interest,
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the rate of interest (in decimal),
- \( t \) is the time in years.

For Mary:
- \( P = 1000 \)
- \( r = 5.5\% = 0.055 \)
- \( t = 3 \) years

Plugging in the values:
\[
I = 1000 \times 0.055 \times 3 = 1000 \times 0.165 = 165
\]

So, **the total interest paid at 5.5% simple interest** would be **$165**.

### (b) Total interest with 3% interest compounded continuously for 3 years

The formula for continuous compounding is:
\[
A = P \times e^{rt}
\]
where:
- \( A \) is the amount of money accumulated after n years, including interest,
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the rate of interest (in decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately 2.71828).

First, let’s find \( A \):
- \( P = 1000 \)
- \( r = 3\% = 0.03 \)
- \( t = 3 \) years

Calculating \( A \):
\[
A = 1000 \times e^{0.03 \times 3}
\]
\[
A = 1000 \times e^{0.09} \approx 1000 \times 1.09417 \approx 1094.17
\]

Now, to find the total interest \( I \), we subtract the principal from the amount accumulated:
\[
I = A - P = 1094.17 - 1000 \approx 94.17
\]

So, **the total interest paid at 3% compounded continuously** would be approximately **$94.17**.

### (c) Comparison of total interests

- Total interest with 5.5% simple interest: **$165**
- Total interest with 3% compounded continuously: **$94.17**

Thus, **the option with 3% compounded continuously results in less total interest**, specifically **$94.17**, compared to **$165** for the simple interest option.