3. (TCO A) The following questions are worth 5 points each. Please show all work.

a. Inflation is expected to average five percent for the long term and Mr. Smith earned $74,000 this year, how much must he earn in 20 years just to keep up with inflation and maintain the balance between his income and his increasing expenditures?

74000*2.653=196,322

b. Jamie wants to have $2,000,000 for her retirement in 35 years. How much should she save annually if she thinks she can earn eight percent on her investments?

2000000*.094=

c. The Flemings will need $160,000 annually for 30 years during retirement. How much will they need at retirement if they can earn a four percent rate of return?

d. The Hamptons want to have $2,500,000 for their retirement in 30 years. How much should they save annually if they think they can earn seven percent on their investments?

To answer these questions, we need to use the concept of future value and present value in finance. Future value represents the value of an amount of money at a future point in time, considering the potential growth of that money. Present value, on the other hand, represents the current value of an amount of money, considering the time value of money.

Here's how you can get the answers for these questions:

a. To calculate how much Mr. Smith must earn in 20 years to keep up with inflation, we need to consider the future value of his current income. Since inflation is expected to average 5% for the long term, we can use the future value formula: Future Value = Present Value * (1 + Inflation Rate)^Number of Years

In this case, Mr. Smith's current income is $74,000. Plugging in the values, we get:

Future Value = $74,000 * (1 + 0.05)^20
Future Value = $74,000 * (1.05)^20
Future Value = $74,000 * 2.653
Future Value = $196,322

So, Mr. Smith must earn approximately $196,322 in 20 years to keep up with inflation.

b. To calculate how much Jamie should save annually for her retirement, we need to consider the future value of her savings. We can use the future value of an ordinary annuity formula: Future Value = Annual Savings * [(1 + Interest Rate)^Number of Years - 1] / Interest Rate

In this case, Jamie wants to have $2,000,000 for her retirement in 35 years, and she expects to earn an 8% interest rate on her investments. Plugging in the values, we get:

$2,000,000 = Annual Savings * [(1 + 0.08)^35 - 1] / 0.08
$2,000,000 = Annual Savings * (1.08^35 - 1) / 0.08

To solve for Annual Savings, we can rearrange the formula:

Annual Savings = $2,000,000 * 0.08 / (1.08^35 - 1)
Annual Savings = $180,000 / (28.246 - 1)
Annual Savings = $180,000 / 27.246
Annual Savings ≈ $6,610.02

So, Jamie should save approximately $6,610.02 annually.

c. To calculate how much the Flemings will need at retirement, we need to consider the present value of their desired annual retirement income. We can use the present value of an annuity formula: Present Value = Annual Income / (1 + Interest Rate)^Number of Years

In this case, the Flemings will need $160,000 annually for 30 years during retirement, and they can earn a 4% rate of return. Plugging in the values, we get:

Present Value = $160,000 / (1 + 0.04)^30
Present Value = $160,000 / (1.04^30)

To solve for the Present Value, we can calculate:

Present Value ≈ $160,000 / 2.208
Present Value ≈ $72,463.77

So, the Flemings will need approximately $72,463.77 at retirement.

d. To calculate how much the Hamptons should save annually for their retirement, we need to consider the future value of their savings. Using the same future value of an ordinary annuity formula as in question b, we can plug in the values:

$2,500,000 = Annual Savings * [(1 + 0.07)^30 - 1] / 0.07

To solve for Annual Savings, we can rearrange the formula:

Annual Savings = $2,500,000 * 0.07 / (1.07^30 - 1)
Annual Savings = $175,000 / (14.972 - 1)
Annual Savings = $175,000 / 13.972
Annual Savings ≈ $12,518.58

So, the Hamptons should save approximately $12,518.58 annually.

I hope this explanation helps you understand how to solve these finance questions.