Question

You invest b0 dollars in an account that draws interest at a monthly rate of r as a decimal, compounded monthly. After t months, the balance in dollars is given by the following equation: b = b0(1 + r)ᵗ. (a) Let a denote the APR as a decimal. Then the monthly rate as a decimal is equal to a divided by 12. Express the balance in terms of the initial investment, the number of months, and the APR as a decimal. b(t) = b0(1 + a/12)ᵗ. (b) Let a denote the APR as a decimal, and let a denote the APR as a percentage. Then a is equal to a divided by 100. Express the balance in terms of the initial investment, the number of months, and the APR as a percentage. b(t) = b0(1 + a/100)ᵗ. (c) Note that y years is equivalent to 12y months. Express the balance in terms of the initial investment, the number y of years, and the APR as a percentage. b(y) = b0(1 + a/100)⁽¹²ʸ⁾.

Answer 1

The student's question pertains to finding the future value of an investment using the compound interest formula, with variations to account for different ways of expressing the APR and the investment duration, either in months or years.

Step-by-step explanation:

The student is dealing with the concept of compound interest, which is a fundamental topic in Mathematics, particularly in financial mathematics. The formulas provided are used to calculate the future value of an investment based on the initial amount b0, the annual percentage rate (APR) either as a decimal or a percentage, and the number of periods the money is invested for, which can be in months or years.

Equations for Compound Interest:

• (a) b(t) = b0(1 + a/12)^t - The balance after t months, for an APR a as a decimal.
• (b) b(t) = b0(1 + a/100)^t - The balance after t months, for an APR a as a percentage.
• (c) b(y) = b0(1 + a/100)^(12y) - The balance after y years, for an APR a as a percentage.

Understanding these equations is crucial for those looking to invest money and for calculating the returns on those investments over different periods of time.