Maria deposits $\$ 40,000$ into an account that pays $4 \%$ interest per year, compounded annually. John deposits $\$ 40,000$ into an account that also pays $4 \%$ per year. But it is simple interest. Find the interest Maria and John earn during each of the first three years. Then decide who earns more interest for each year. Assume there are no withdrawals and no additional deposits.
Solution
Step 1 ：Maria and John both deposit $40,000 into accounts that pay 4% interest per year. Maria's interest is compounded annually, while John's is simple.
Step 2 ：For Maria, the interest earned each year is added to the principal, so the amount on which the interest is calculated increases each year. The formula for compound interest is \(A = P(1 + r/n)^{nt}\), 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), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years. In this case, \(P = \$40,000\), \(r = 4% = 0.04\), \(n = 1\) (since the interest is compounded annually), and \(t = 1\) year for the first year, \(2\) years for the second year, and \(3\) years for the third year.
Step 3 ：For John, the interest earned each year is the same because it's based on the original principal. The formula for simple interest is \(I = Prt\), where \(I\) is the interest, \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is the time the money is invested for. In this case, \(P = \$40,000\), \(r = 4% = 0.04\), and \(t = 1\) year for each of the three years.
Step 4 ：Maria earns \$1600 in the first year, \$3264 in the second year, and \$4994.56 in the third year. John earns \$1600 each year.
Step 5 ：So, Maria earns more interest than John in the second and third years, but they earn the same amount of interest in the first year.
Step 6 ：\(\boxed{\text{Final Answer: Maria and John both earn \$1600 in the first year. In the second year, Maria earns \$3264 and John earns \$1600. In the third year, Maria earns \$4994.56 and John earns \$1600. So, Maria earns more interest than John in the second and third years.}}\)
