Questions: Determine the present value, P, you must invest to have the future value, A, at simple interest rate r after time t. Round answer to the nearest dollar. A = 146.30, r = 11 %, t = 3 years A. 117 B. 143 C. 113 D. 110

Determine the present value, P, you must invest to have the future value, A, at simple interest rate r after time t. Round answer to the nearest dollar. A = 146.30, r = 11 %, t = 3 years A. 117 B. 143 C. 113 D. 110
Transcript text: Determine the present value, $P$, you must invest to have the future value, $A$, at simple interest rate $r$ after time $t$. Round answer to the nearest dollar. \[ A=\$ 146.30, r=11 \%, t=3 \text { years } \] A. $\$ 117$ B. $\$ 143$ C. $\$ 113$ D. $\$ 110$
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Solution

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Solution Steps

To determine the present value \( P \) that must be invested to achieve a future value \( A \) at a simple interest rate \( r \) after time \( t \), we use the formula for simple interest:

\[ A = P(1 + rt) \]

Rearrange this formula to solve for \( P \):

\[ P = \frac{A}{1 + rt} \]

Substitute the given values into the formula to calculate \( P \).

Step 1: Identify the Given Values

We are given the following values:

  • Future value \( A = 146.30 \)
  • Interest rate \( r = 11\% = 0.11 \)
  • Time \( t = 3 \) years
Step 2: Use the Simple Interest Formula

The formula for future value in terms of present value is given by: \[ A = P(1 + rt) \] To find the present value \( P \), we rearrange the formula: \[ P = \frac{A}{1 + rt} \]

Step 3: Substitute the Values

Substituting the known values into the formula: \[ P = \frac{146.30}{1 + (0.11 \times 3)} \] Calculating the denominator: \[ 1 + (0.11 \times 3) = 1 + 0.33 = 1.33 \] Now substituting back: \[ P = \frac{146.30}{1.33} \]

Step 4: Calculate Present Value

Calculating \( P \): \[ P \approx 110.00 \] Rounding to the nearest dollar gives: \[ P \approx 110 \]

Final Answer

The present value \( P \) that must be invested is \\(\boxed{110}\\).

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