The present value $(P V)$ 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, $P V$ , to be invested in a savings account earning $5 \%$ interest compounded annually for $t$ years? $\newline$ Choose $1$ answer: $\newline$ (A) $P V(t)=10,000(1.05)^{t}$ $\newline$ (B) $P V(t)=10,000(1.05)^{-t}$ $\newline$ (C) $P V(t)=10,000(1+0.05 t)$ $\newline$ (D) $P V(t)=10,000(1-0.05 t)$

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Grade 8

Interpreting Linear Expressions

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Q. The present value

(P V)

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,

P V

, to be invested in a savings account earning

5 \%

interest compounded annually for

t

years?

\newline

Choose

1

answer:

\newline

(A)

P V(t)=10,000(1.05)^{t}

\newline

(B)

P V(t)=10,000(1.05)^{-t}

\newline

(C)

P V(t)=10,000(1+0.05 t)

\newline

(D)

P V(t)=10,000(1-0.05 t)

Reverse Compound Interest Formula: To find the present value, we need to use the formula for compound interest in reverse, which is $PV = \frac{FV}{(1 + r)^t}$ , where $FV$ is the future value, $r$ is the interest rate, and $t$ is the time in years.
Given Values: We are given $FV = \$10,000$ , $r = 5\%$ or $0.05$ , and $t$ is the number of years. We need to find the function that represents $PV$ .
Substitute Values: Substitute the given values into the formula: $PV = \frac{10,000}{(1 + 0.05)^t}.$
Simplify Equation: This simplifies to $PV = \frac{10,000}{(1.05)^t}$ , which is the same as $PV = 10,000 \times (1.05)^{-t}$ .
Correct Present Value Function: Looking at the choices, the correct function that models the present value is $PV(t) = 10,000 \times (1.05)^{-t}$ .