Question

The amount of an investment will double in approximately $70/p$ years, where $p$ is the percent interest, compounded annually. If Thelma invests $\mathrm{\$}40,000$ in a long-term $\mathrm{CD}$ that pays 5 percent interest, compounded annually, what will be the approximate total value of the investment when Thelma is ready to retire 42 years later? A $\mathrm{\$}280,000$ B $\mathrm{\$}320,000$ C $\mathrm{\$}360,000$ D $\mathrm{\$}450,000$ E $\mathrm{\$}540,000$

Short answer

The total value of the investment will be approximately $320,000.

Step-by-step solution

- Calculate the Doubling Time

Using the formula for the doubling time, \ $$\text{Doubling Time}=\frac{70}{p}$$ for an interest rate of 5%, \ $$\text{Doubling Time}=\frac{70}{5}=14\text{ years}$$

- Determine Number of Doublings in 42 Years

The investment period is 42 years. Divide this by the doubling time to find out how many times the investment doubles: \ $$\text{Number of Doublings}=\frac{42}{14}=3$$

- Calculate the Final Amount

If the investment doubles 3 times starting from $\mathrm{\$}40,000$, then: \ $$\mathrm{\$}40,000\times 2^3=\mathrm{\$}40,000\times 8=\mathrm{\$}320,000$$

Key concepts

compound interest

Compound interest is interest calculated on the initial principal, which also includes all the accumulated interest from previous periods. This means that the investment grows exponentially rather than linearly. The formula to calculate compound interest is:

$$A=P(1+\frac{r}{n}{)}^{nt}$$

where:

• A: the amount of money accumulated after n years, including interest.
• P: the principal amount (the initial amount of money).
• r: the annual interest rate (in decimal form).
• n: the number of times that interest is compounded per year.
• t: the number of years the money is invested for.

In the context of Thelma's investment, the interest is compounded annually, so the value of n is 1.

doubling time formula

The doubling time formula is a quick and easy way to estimate how long it takes for an investment to double. The rule of 70 is often used to determine this:

$$\text{Doubling Time}=\frac{70}{p}$$

where p is the annual interest rate as a percentage. For Thelma’s investment at 5% interest, compounded annually:
$$\text{Doubling Time}=\frac{70}{5}=14\text{ years}$$

This tells us that her investment will double every 14 years.

retirement savings

Planning for retirement savings is essential for securing your financial future. Estimating the future value of investments helps individuals to understand how much they will have when they retire. For Thelma, who invests $40,000overaperiodof42years,i{t}^{\prime }scrucialtousethedoublingtimeformulaandcompoundinterestprinciplestoforeseeherinvestmentgrowth.Giventhecalculatedvalues,Thelm{a}^{\prime }sinitialinvestmentwouldgrowto$320,000 when she retires in 42 years, thanks to the compound interest earned over multiple periods.

long-term investment

Long-term investments involve committing money for an extended period, often several decades, to reap the benefits of growth through interest or asset appreciation. Long-term investments like Thelma's are ideal for retirement savings due to the exponential growth rendered by compound interest. Such investments require patience, as the true benefits are realized over years or decades. The key advantage of long-term investments lies in their ability to multiply the invested funds multiple times, as demonstrated in Thelma's investment doubling three times to reach $320,000.

percent interest calculation

Calculating percent interest is fundamental in determining how investments grow over time. The annual interest rate is expressed as a percentage and applied to calculate how much interest one earns each year. For Thelma's investment at 5% interest, the percent interest calculation is essential to understanding how the investment doubles. This 5% rate annually compounds, meaning that each year's interest is calculated on the new total, including the previously earned interest. Therefore, consistent calculation and compounding significantly increase the investment value over the years.