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:
\(\[ A = P(1 + r)^t \]\)
where:
- \(\( A \)\) is the amount of money accumulated after n years, including interest.
- \(\( P \)\) is the principal amount (the initial amount of money).
- \(\( r \)\) is the annual interest rate (in decimal form).
- \(\( t \)\) is the time the money is invested for, in years.
Given:
- The final amount (\(\( A \)\)) that we want is K10,000.
- The annual interest rate (\(\( r \)\)) is 3.5%, which we can write in decimal form as 0.035.
- The time (\(\( t \)\)) is 6 years.
We need to calculate the principal amount (\(\( P \)\)), 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 \(\( P \)\):
\(\[ P = \frac{A}{(1 + r)^t} \]\)
Substitute in the given values:
\(\[ P = \frac{10,000}{(1 + 0.035)^6} \]\)
Calculate the value inside the parentheses:
\(\[ 1 + 0.035 = 1.035 \]\)
Raise this value to the sixth power (since the investment is over 6 years):
\(\[ (1.035)^6 \]\)
Now, divide the final amount (\(\( A \)\)) by this result to find the principal (\(\( P \)\)):
\(\[ P = \frac{10,000}{(1.035)^6} \]\)
Calculate this expression to determine the principal amount (\(\( P \)\)) needed:
\(\[ P ≈ 8135.006443077529 \]\)
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.
