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Chapter 3: Problem 5

Barbara wants to restore her 66 Mustang in 4 years. She puts \(\$ 200\) into an account every month that pays 4.5\(\%\) interest, compounded monthly. How much is in the account after 4 years?

Short Answer

Expert verified
After calculating the formula, the amount in the account after 4 years will be about \$10,960. This is the final answer after considering monthly deposits and compounded interest.

Step by step solution

01

Identify Known Values

Firstly, identify all the known values: the monthly deposit (\$200), monthly interest rate (4.5\% per year, so \(0.045 / 12\) per month), and the total time frame (4 years or 48 months)
02

Apply Future Value of a Series Formula

The future value of a series formula (for regular deposits) is given by \(FV = P \times \frac{{(1 + r)^n - 1}}{r}\), where P is the regular deposit, r is the interest rate per period, and n is the number of periods. Plugging in the known values, we find \(FV = 200 \times \frac{{(1 + 0.045/12)^{48} - 1}}{0.045/12}\)
03

Calculate the Future Value

Finally, calculate the numerical value of the future value which will show how much will be in the account after 4 years.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compounded Interest
When it comes to maximizing savings, understanding the concept of compounded interest is crucial. Compounded interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. This means that interest is earned not only on your original amount but also on the interest that has been added over time. In Barbara's case, where she's saving for her Mustang restoration, the interest earned each month will be added to the total balance, and in the following month, she will earn interest on this new amount.

The formula used to calculate the compounded future value is quite standard:
\[ FV = P \times \frac{{(1 + r)^n - 1}}{r} \]
It takes into account the amount you're depositing each period (\(P\)), the interest rate per period (\(r\)), and the number of periods interest is compounded (\(n\)).
Monthly Deposits
Regular monthly deposits are a strategic approach to build savings over time. By consistently depositing a set amount of money, like Barbara's \($200\) every month, you take advantage of compounded interest and can significantly grow your savings. It's this consistent action of savings that can help reach financial goals, such as Barbara's desire to restore her classic car.

When these deposits earn compounded interest, as in Barbara's case, each deposit begins to accumulate its own 'earning power', contributing to an accelerating balance over time. This is often referred to as 'the snowball effect', as each deposit adds more mass to your growing snowball of savings.
Savings Plan Mathematics
Savings plan mathematics involves using formulas and calculations to determine how much your savings will amount to over a specific period, factoring in contributions and compounded interest. It is an essential part of personal finance planning, helping savers like Barbara to predict future savings based on mathematics rather than guesswork.

To maximize the benefits of any savings plan, it is important to understand variables like deposit frequency, consistent deposit amounts, interest rates, and the compounding period. Barbara's 4-year savings plan for her Mustang restoration expertly showcases how savings plan mathematics work: by making steady monthly deposits, the compounded interest grows her savings substantially, allowing her to reach her goal within her set timeframe.

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Most popular questions from this chapter

Joanne deposits \(\$ 4,300\) into a one-year \(\mathrm{CD}\) at a rate of \(4.3 \%,\) compounded daily. a. What is her ending balance after the year? b. How much interest does she earn? c. What is her annual percentage yield to the nearest hundredth of a percent?

Mike deposits \(\$ 5,000\) in a three-year \(\mathrm{CD}\) account that yields 3.5\(\%\) interest, compounded weekly. What is his ending balance at the end of three years?

Bob wants \(\$ 50,000\) at the end of 7 years in order to buy a car. If his bank pays 4.2\(\%\) interest, compounded annually, how much must he deposit each year in order to reach his goal?

Albert Einstein said that compound interest was 鈥�. . .the most powerful thing I have ever witnessed.鈥� Work through the following exercises to discover a pattern Einstein discovered which is now known as the Rule of 72.. a. Suppose that you invest \(\$ 2,000\) at a 1\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? b. Suppose that you invest \(\$ 4,000\) at a 2\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? c. Suppose that you invest \(\$ 20,000\) at a 6\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? d. Albert Einstein noticed a very interesting pattern when an initial deposit doubles. In each of the three examples above, multiply the value of t that you determined times the percentage amount. For example, in a. multiply t by 1. What do you notice? e. Einstein called this the Rule of 72 because for any initial deposit and for any interest percentage, \(72 \div\) (percentage) will give you the approximate number of years it will take for the initial deposit to double in value. Einstein also said that 鈥淚f people really understood the Rule of 72 they would never put their money in banks.鈥� Suppose that a 10-year-old has $500 to invest. She puts it in her savings account that has a 1.75% annual interest rate. How old will she be when the money doubles?

Grandpa Joe wants to open an account for his grandchildren that he hopes will have \(\$ 80,000\) in it after 20 years. How much must he deposit now into an account that yields 2.75\(\%\) interest, compounded monthly, so he can be assured of reaching his goal?
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