Multiple choice

$\displaystyle 1,000 \left( 1 + \frac{r}{1,200}\right)^{12}$ The expression above gives the amount of money, in dollars, generated in a year by a $\$1,000$ deposit in a bank account that pays an annual interest rate of $r\%$, compounded monthly. Which of the following expressions shows how much additional money is generated at an interest rate of $5\%$ than at an interest rate of $3\%$?

  1. $\displaystyle 1,000 \left(1 + \frac{5-3}{1,200}\right)^{12}$
  2. $\displaystyle 1,000 \left(1 + \frac{\displaystyle \frac{5}{3}}{1,200}\right)^{12}$
  3. $\displaystyle \frac{1,000 \left(1 + \displaystyle \frac{5}{1,200}\right)^{12}}{1,000 \left(1 + \displaystyle \frac{3}{1,200}\right)^{12}}$
  4. $\displaystyle 1,000 \left(1 + \displaystyle \frac{5}{1,200}\right)^{12} - \displaystyle 1,000 \left(1 + \displaystyle \frac{3}{1,200}\right)^{12}$
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D Correct answer
Explanation

The amount generated is A(r) = 1000 * (1 + r/1200)^12. Additional money at 5% vs 3% is A(5) - A(3).

AI explanation

To find the additional money generated, calculate the final amounts for both interest rates and subtract them. The amount generated at 5% is 1000 * (1 + 5/1200)^12 and the amount generated at 3% is 1000 * (1 + 3/1200)^12. The difference between these two amounts is 1000 * (1 + 5/1200)^12 - 1000 * (1 + 3/1200)^12.

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