23:18 |12 siyavula.com Siy \( 2 \sqrt{u} / 2 \) The required simple interest rate is \( \square \) 11,11 \( \% \) per annum. Boo-yah! That's correct. Show the full solution 2. Calculate the required annual interest rate to achieve this growth, if the investment earned compound interest (compounded annually). INSTRUCTIONS: - Round your answer to two decimal places, if necessary. - You do not need to type the percentage symbol, as it has already been typed for you.

To calculate the required annual interest rate for an investment that earned compound interest, we can use the formula for compound interest: \[ A = P(1 + r)^n \] 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 (decimal). - \( n \) is the number of years the money is invested or borrowed. ### Step 1: Identify the known values From the problem, we know: - The final amount \( A \) is not explicitly given, but we can assume it is the amount that corresponds to the simple interest rate of \( 11.11\% \) per annum. - The principal amount \( P \) is also not given, but we can assume it is \( 100 \) for simplicity. - The time period \( n \) is not specified, but we can assume it is \( 1 \) year for the sake of calculating the annual interest rate. ### Step 2: Calculate the final amount using simple interest The formula for simple interest is: \[ I = P \times r \times n \] Where: - \( I \) is the interest earned. - \( r \) is the simple interest rate (as a decimal). - \( n \) is the time in years. Using \( P = 100 \), \( r = 0.1111 \), and \( n = 1 \): \[ I = 100 \times 0.1111 \times 1 = 11.11 \] Thus, the final amount \( A \) after 1 year is: \[ A = P + I = 100 + 11.11 = 111.11 \] ### Step 3: Set up the compound interest equation Now we can set up the compound interest equation with the values we have: \[ 111.11 = 100(1 + r)^1 \] ### Step 4: Solve for \( r \) To find \( r \), we can rearrange the equation: \[ 1 + r = \frac{111.11}{100} \] Calculating the right side: \[ 1 + r = 1.1111 \] Now, subtract 1 from both sides: \[ r = 1.1111 - 1 = 0.1111 \] ### Step 5: Convert \( r \) to a percentage To convert \( r \) to a percentage, we multiply by 100: \[ r = 0.1111 \times 100 = 11.11\% \] ### Conclusion The required annual interest rate to achieve this growth, if the investment earned compound interest (compounded annually), is: \[ \boxed{11.11} \] This confirms that the compound interest rate is the same as the simple interest rate in this case, given the assumptions made.