The present value $PV$ of an investment is the amount that should be invested today at a specified interest rate in order to earn a certain amount at a future date. The amount desired is called the future value. For a future value of $\$10,000$ , which of the following functions models the present value, $PV$ , to be invested in a savings account earning $5\%$ interest compounded annually for $t$ years?

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Grade 7

Compound interest

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Q. The present value

PV

of an investment is the amount that should be invested today at a specified interest rate in order to earn a certain amount at a future date. The amount desired is called the future value. For a future value of

\$10,000

, which of the following functions models the present value,

PV

, to be invested in a savings account earning

5\%

interest compounded annually for

t

years?

Identify Given Values: We need to use the formula for the present value (PV) of an investment with compound interest. The formula is: $\newline$ $PV = \frac{FV}{(1 + r)^t}$ $\newline$ where $FV$ is the future value, $r$ is the annual interest rate (as a decimal), and $t$ is the number of years.
Substitute Values into Formula: First, we identify the given values: $\newline$ $FV = \$10,000$ (future value) $\newline$ $r = 5\%$ or $0.05$ (annual interest rate) $\newline$ $t =$ unknown (number of years) $\newline$ We will express the present value $PV$ as a function of $t$ .
Simplify Expression: Now, we substitute the given values into the formula: $\newline$ $PV = \frac{10000}{(1 + 0.05)^t}$ $\newline$ This function will give us the present value for any number of years $t$ .
Final Present Value Function: We simplify the expression to make it clear: $\newline$ $PV = \frac{10000}{(1.05)^t}$ $\newline$ This is the function that models the present value.