To calculate the amount that must be invested at an annual compound interest rate of 3.5% in order to have K10,000 at the end of 6 years, we can use the compound interest formula:
[tex]\[ A = P(1 + r)^t \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Given:
- The final amount ([tex]\( A \)[/tex]) that we want is K10,000.
- The annual interest rate ([tex]\( r \)[/tex]) is 3.5%, which we can write in decimal form as 0.035.
- The time ([tex]\( t \)[/tex]) is 6 years.
We need to calculate the principal amount ([tex]\( P \)[/tex]), which is the amount that needs to be invested now to get K10,000 after 6 years.
Rearrange the compound interest formula to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{A}{(1 + r)^t} \][/tex]
Substitute in the given values:
[tex]\[ P = \frac{10,000}{(1 + 0.035)^6} \][/tex]
Calculate the value inside the parentheses:
[tex]\[ 1 + 0.035 = 1.035 \][/tex]
Raise this value to the sixth power (since the investment is over 6 years):
[tex]\[ (1.035)^6 \][/tex]
Now, divide the final amount ([tex]\( A \)[/tex]) by this result to find the principal ([tex]\( P \)[/tex]):
[tex]\[ P = \frac{10,000}{(1.035)^6} \][/tex]
Calculate this expression to determine the principal amount ([tex]\( P \)[/tex]) needed:
[tex]\[ P ≈ 8135.006443077529 \][/tex]
So, an investment of approximately K8,135.01 at an annual interest rate of 3.5%, compounded annually, is needed to yield K10,000 at the end of 6 years.
