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?

1 answer

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.