Question

If John invested $\mathrm{\$}1$ at 5 percent interest compounded annually, the total value of the investment, in dollars, at the end of 4 years would be A $$(1.5{)}^4$$ B $$4(1.5)$$ C $$(1.05{)}^4$$ D $$1+(0.05{)}^4$$ E $$1+4(0.05)$$

Short answer

Option C: $(1.05{)}^4$

Step-by-step solution

- Understand Compound Interest Formula

The formula for compound interest is given by $$A=P(1+r/n{)}^{nt}$$where:- $A$ is the amount of money accumulated after n years, including interest.- $P$ is the principal amount (initial investment).- $r$ is the annual interest rate (decimal).- $n$ is the number of times that interest is compounded per year.- $t$ is the time the money is invested for in years.

- Plug in the Given Values

In this problem, $P=1$ (since John invested $1), $r=0.05$ (since the interest rate is 5 percent), $n=1$ (compounded annually), and $t=4$ (for 4 years). Plug these values into the formula.$$A=1(1+0.05/1{)}^{1\ast 4}$$

- Simplify Inside the Parentheses

First, simplify inside the parentheses:$$1+0.05/1=1.05$$ So, the formula becomes:$$A=1(1.05{)}^4$$

- Identify the Correct Option

Now we match the simplified expression $1(1.05{)}^4$ with the given answer choices:- A $(1.5{)}^4$- B $4(1.5)$- C $(1.05{)}^4$- D $1+(0.05{)}^4$- E $1+4(0.05)$The correct choice is clearly option C.

- Final Answer

The total value of the investment at the end of 4 years is given by option C: $(1.05{)}^4$.

Key concepts

Investing

When we talk about investing, it means putting money into something with the expectation that it will grow over time. It could be in stocks, bonds, real estate, or a simple savings account. In John’s case, he invested a small amount, $\mathrm{\$}1$, hoping it would grow over four years. The goal of investing is to generate additional income or achieve capital gains. It’s important to carefully choose where to invest. You want to make sure that your investment grows and that you understand the risks involved.

Basic principles of investing include:

• Risk and return: Taking higher risks may bring higher returns.
• Diversification: Spreading your investments can reduce risk.
• Compounding: Reinvesting earnings can increase returns.

Understanding these can help you make smart investment decisions and grow your wealth over time.

Annual Interest Rate

The annual interest rate is the percentage of the principal amount that is paid as interest over one year. In John’s case, the annual interest rate is 5%, written as $r=0.05$ in decimal form for calculation purposes. This rate influences how much money you will earn on your investment over time. Higher annual interest rates mean your money grows faster.

Here are a few key points about annual interest rates:

• Expressed as a percentage of the investment.
• Determines the growth rate of an investment.
• Annual interest rates can be fixed or variable.

It’s important to understand the interest rate associated with your investment because it directly impacts how much your investment will grow.

Financial Mathematics

Financial mathematics is the application of mathematical methods to financial problems. It includes calculations like compound interest, which helps us understand how investments grow over time. In the example we solved, we used a specific formula to calculate the future value of John’s investment.

The compound interest formula 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 (initial investment).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested for in years.

Understanding financial mathematics allows us to make better decisions regarding investments and savings by computing future values accurately.

Interest Compounding

Interest compounding refers to the process where interest earned on an investment is reinvested to earn more interest. In John’s case, the interest is compounded annually. This means that at the end of each year, any interest earned is added to the principal amount, and this new total earns interest in subsequent years.

Key points about interest compounding:

• The frequency of compounding can vary (annually, semi-annually, quarterly, etc.).
• More frequent compounding periods can result in higher returns.
• The compound interest formula includes the compounding frequency.

Think of compounding as a snowball effect where the growth of the investment accelerates over time because each period’s interest is calculated on a progressively larger amount. The earlier you start investing, the more you can benefit from compounding.