Benjamin invests money in a bank account which gathers compound interest each year. After 2 years there is 658.20 in the account. After 5 years there is 710.89 in the account. Work out the annual interest rate of the bank account. Give your answer as a percentage to 1 d,p.

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Benjamin invests money in a bank account which gathers compound interest each year. After 2 years there is 658.20 in the account. After 5 years there is 710.89 in the account. Work out the annual interest rate of the bank account. Give your answer as a percentage to 1 d,p.

Answer 1

# Explanation: ## Step 1: We know the formula for compound interest, which is given as: ### \( A = P \times (1 + \frac{r}{n})^{nt} \) Where, A is the final amount, P is the principal amount (the initial amount), r is the annual interest rate, n is the number of compounding periods per year, and t is the time period in years. Since in this case, Ben's money accumulated annually, \( n \) will be equal to 1. ## Steps 2 & 3: Let \( P \) be the initial investment amount, \( r \) the annual interest rate, this gives us two different versions of the interest formula: ### \( 658.20 = P \times (1 + r) ^ 2 \) ### \( 710.89 = P \times (1 + r) ^ 5 \) Unfortunately, these equations have two variables, \( P \) and \( r \), and we cannot solve them as is. ## Step 4: Instead of trying to solve for two variables, we will divide out \( P \) from both equations. This will give us this formula: ### \( \frac{658.20}{710.89} = \frac{(1 + r) ^ 2}{(1 + r) ^ 5} \) ## Step 5: Once simplifying further, the equation becomes: ### \( \frac{658.20}{710.89} = (1 + r)^{-3} \) ## Step 6: By swapping sides and taking the (1/3)-th power and subtracting 1, we can infer the value of \( r \). # Answer: Ben's account has an annual interest rate of recovering the percentage %( Approximately recover `1` d.p.) and provide it as answer ensuring that the calculation amounts are accurate and correctly colored according to the placeholders very precisely.

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Benjamin invests money in a bank account which gathers compound interest each year. After 2 years there is 658.20 in the account. After 5 years there is 710.89 in the account. Work out the annual interest rate of the bank account. Give your answer as a percentage to 1 d,p.

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