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Chapter 6: Problem 1

Use the accompanying table to estimate the number of years it would take \(\$ 100\) to become \(\$ 300\) at the following interest rates compounded annually. a. \(3 \%\) b. \(7 \%\) Compound Interest over 40 Years $$ \begin{array}{ccc} \hline \text { Number of } & \text { Value of } \$ 100 \text { at } & \text { Value of } \$ 100 \text { at } \\ \text { Years } & 3 \%(\$) & 7 \%(\$) \\ \hline 0 & 100 & 100 \\ 10 & 134 & 197 \\ 20 & 181 & 387 \\ 30 & 243 & 761 \\ 40 & 326 & 1497 \\ \hline \end{array} $$

Short Answer

Expert verified
3%: 40 years, 7%: 20 years.

Step by step solution

01

- Identify the target value

We need to find out how many years it will take for an initial investment of \(100 to grow to \)300 at the specified interest rates.
02

- Examine the table for 3% interest

Identify the value in the table that first exceeds \(300 in the 3% column. The values are: - 0 years: \)100 - 10 years: \(134 - 20 years: \)181 - 30 years: \(243 - 40 years: \)326 From the table, \(300 is first exceeded at 40 years when the value reaches \)326.
03

- Examine the table for 7% interest

Identify the value in the table that first exceeds \(300 in the 7% column. The values are: - 0 years: \)100 - 10 years: \(197 - 20 years: \)387 From the table, \(300 is first exceeded at 20 years when the value reaches \)387.
04

- Conclude the number of years

Summarize the results obtained from the previous steps: - At 3%, it takes 40 years to exceed \(300. - At 7%, it takes 20 years to exceed \)300.

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Key Concepts

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

interest rates
Interest rates play a crucial role in how fast your investment grows. The interest rate represents the percentage at which your investment increases over a certain period. Higher interest rates result in faster growth of your investment, while lower interest rates mean slower growth. For example, in the exercise, a 3% interest rate lead to an investment of \(100 growing to \)326 in 40 years, while a 7% interest rate lead to the same \(100 growing to \)387 in just 20 years.
The formula to calculate compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
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 in years.
investment growth
Investment growth is the increase in value of an initial sum of money over time, driven primarily by earning interest. The compounding effect means that the reinvested earnings also generate returns, boosting the total investment value further. In our exercise, we see that the initial \(100 grows significantly over different periods, depending on the interest rate applied. Such growth is essential to understand if you're planning for long-term financial goals like retirement or major purchases.
Consider the table given:
- At 3% interest, \)100 grows to \(326 in 40 years.
- At 7% interest, \)100 grows to \(387 in just 20 years and even reaches \)1497 after 40 years.
Thus, higher interest rates and longer investment periods substantially increase the final value of your investment.
time value of money
The time value of money (TVM) is a financial principle stating that a dollar today is worth more than a dollar in the future. This concept hinges on the idea that money available now can be invested to earn returns, while sums received in the future are subject to risks and inflation. In our example, the different durations required to turn \(100 into \)300 highlight this principle.
- At a lower interest rate (3%), it would take 40 years to exceed $300.
- At a higher interest rate (7%), it only takes 20 years.
Understanding TVM helps in making informed investment decisions, ensuring that funds are invested in the most beneficial way to maximize returns over time.

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Most popular questions from this chapter

a. Generate a table of values to estimate the half-life of a substance that decays according to the function \(y=100(0.8)^{x},\) where \(x\) is the number of time periods, each time period is 12 hours, and \(y\) is in grams. b. How long will it be before there is less than 1 gram of the substance remaining?

(Requires a graphing program.) On the same grid graph \(y_{1}=\ln (x), y_{2}=-\ln (x), y_{3}=\ln (-x)\) and \(y_{4}=-\ln (-x)\) a. Which pairs of function graphs are mirror images across the \(y\) -axis? b. Which pairs of function graphs are mirror images across the \(x\) -axis? c. What predictions would you make about the graphs of the functions \(f(x)=a \ln x\) and \(g(x)=-a \ln x\) ? Using technology, test your predictions for different values of \(a\) d. What predictions would you make about the graphs of the functions \(f(x)=a \ln x\) and \(g(x)=a \ln (-x) ?\) Using technology, test your predictions for different values of \(a\)

Expand each logarithm using only the numbers \(2,3, \log 2,\) and \(\log 3\). a. \(\log 9\) b. \(\log 18\) c. \(\log 54\)

Wikipedia is a popular, free online encyclopedia (at \(e n .\) wikipedia.org) that anyone can edit. (So articles should be taken with "a grain of salt.") One Wikipedia article claims that the number of articles posted on Wikipedia has been growing exponentially since October 23,2002 . At that date there were approximately 90 thousand articles posted, and the growth rate was about \(0.2 \%\) per day. a. Create an exponential model for the growth in Wikipedia articles. b. What is the doubling time? Interpret your answer.

Match each exponential function in parts (a)-(d) with its logarithmic form in parts (e)-(h). a. \(y=10,000(2)^{x}\) b. \(y=1000(1.4)^{x}\) c. \(y=\left(3 \cdot 10^{6}\right)(0.8)^{x}\) d. \(y=1000(0.45)^{x}\) e. \(\log y=6.477-0.097 x\) f. \(\log y=4+0.301 x\) g. \(\log y=3-0.347 x\) h. \(\log y=3+0.146 x\)
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