Question

Sean currently has an account balance of $3,981.04. He opened the account five years ago with a deposit of $3,760.19. If the interest compounds daily, what is the interest rate on the account? (3 points) 1.1% 0.6% 4.6% 5.6%

Answer 1

`\text{The amount in an account compounding interest rate is given by}\\ \\ A=P\left ( 1+(r)/(n) \right )^(nt)\\ \\ \text{here A is amount after t year, P is the principal amount, r is interest}\\ \text{rate in decimals, n is the number of times compounded in a year.}`

here we have given that currently amount is A=$ 3981.04

the account opened with a principle amount, P=$ 3760.19

`\text{ the interest compounded daily, so }n=365\\ \text{and the time is t=5 years. so let the interest rate is r. so}\\ \\ A=P\left ( 1+(r)/(n) \right )^(nt)\\ \\ 3981.04=3760.19\left ( 1+(r)/(365) \right )^(365(5))`

`\Rightarrow (3981.04)/(3760.19)=\left ( 1+(r)/(365) \right )^(1825)\\ \\ \Rightarrow 1.0587=\left ( 1+(r)/(365) \right )^(1825)\\ \\ \Rightarrow \left ( 1+(r)/(365) \right )^(1825)=1.0587\\ \\ \Rightarrow \left ( 1+(r)/(365) \right )=(1.0587)^{(1)/(1825)}`
`\Rightarrow 1+(r)/(365) \approx 1.0000313\\ \\ \Rightarrow (r)/(365)=1.0000313-1\\ \\ \Rightarrow (r)/(365)=0.0000313\\ \\ \Rightarrow r=0.0000313* 365\\ \\ \Rightarrow r\approx 0.011`

Hence the interest rate on the account is: 1.1%