Determine the present value \(P\) that must be invested to have the future value \(A\) at a simple interest rate \(r\) after time \(t\).

Given:
[tex]\[ A = \$5000, \quad r = 1.55\%, \quad t = 6 \text{ months} \][/tex]



Answer :

To determine the present value \( P \), we need to use the formula for calculating the present value in a situation involving simple interest. The formula is:

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

where:
- \( A \) is the future value you want to achieve.
- \( r \) is the simple interest rate (expressed as a decimal).
- \( t \) is the time the money is invested for, in years.

Given the values:
- \( A = \$5000 \)
- \( r = 1.55\% = 0.0155 \) (as a decimal)
- \( t = 6 \) months, which needs to be converted into years. Since there are 12 months in a year, \( t = \frac{6}{12} = 0.5 \) years.

We substitute these values into the formula:

[tex]\[ P = \frac{5000}{1 + (0.0155 \times 0.5)} \][/tex]

First, calculate the term in the denominator:

[tex]\[ 1 + (0.0155 \times 0.5) = 1 + 0.00775 = 1.00775 \][/tex]

Now divide the future value by this result:

[tex]\[ P = \frac{5000}{1.00775} \approx 4961.55 \][/tex]

Hence, the present value \( P \) that must be invested to achieve the future value of \$5000 after 6 months at a simple interest rate of 1.55% is approximately:

[tex]\[ P \approx \$4961.55 \][/tex]