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Chapter 5: Problem 24

A bank compounds interest annually at \(4 \%\). a. Write an equation for the value \(V\) of \(\$ 100\) in \(t\) years. b. Write an equation for the value \(V\) of \(\$ 1000\) in \(t\) years. c. After 20 years will the total interest earned on \(\$ 1000\) be ten times the total interest earned on \(\$ 100 ?\) Why or why not?

Short Answer

Expert verified
The equations are \( V = 100(1.04)^t \) and \( V = 1000(1.04)^t \). After 20 years, the interest earned on \$1000 is approximately ten times that on \$100 because the interest rates are compounded equally.

Step by step solution

01

- Understanding the compound interest formula

The compound interest formula used to calculate the future value with annual compounding is given by: \[ V = P(1 + r)^t \] where:\( V \) is the future value of the investment after\( t \) years, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (decimal form), and \( t \) is the number of years.
02

- Equation for part (a)

For part (a), the principal amount \( P \) is \$100 and the interest rate \( r \) is 0.04. Plug these values into the compound interest formula: \[ V = 100(1 + 0.04)^t \] Simplifying further: \[ V = 100(1.04)^t \]
03

- Equation for part (b)

For part (b), the principal amount \( P \) is \$1000 and the interest rate \( r \) is 0.04. Plug these values into the compound interest formula: \[ V = 1000(1 + 0.04)^t \] Simplifying further: \[ V = 1000(1.04)^t \]
04

- Comparing the interest earned after 20 years

To solve part (c), calculate the interest earned on \$100 and \$1000 after 20 years. \[ V_{100} = 100(1.04)^{20} \] \[ V_{1000} = 1000(1.04)^{20} \] The interest earned is the future value minus the principal amount. Calculate \( V_{100} \) and \( V_{1000} \) first: \[ V_{100} = 100(1.04)^{20} \approx 219.11 \] \[ V_{1000} = 1000(1.04)^{20} \approx 2191.12 \] The interest earned on \$100 is \( 219.11 - 100 = 119.11 \). The interest earned on \$1000 is \( 2191.12 - 1000 = 1191.12 \). Lastly, compare the ratios of interest: \[ 1191.12 / 119.11 \approx 10 \] The total interest earned on \$1000 is approximately ten times the total interest earned on \$100 because the bank compounds interest at the same rate for both.

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Key Concepts

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

Future Value Calculation
The concept of future value is essential when discussing investments and finances. It represents how much an investment made today will be worth in the future, considering interest earned over time. To calculate future value with compound interest, we use the formula: \( V = P(1 + r)^t \). Here: \begin{itemize} \item V stands for the future value. \item P represents the principal or initial investment. \item r is the annual interest rate (expressed as a decimal). \item t denotes the number of years the money is invested or borrowed. \end{itemize} Plugging the values of principal, interest rate, and time period into this formula gives us the future worth of the investment. Whether you are saving for a house, education, or retirement, this formula helps you understand the growth of your money over time.
Annual Interest Rate
The annual interest rate is a crucial factor in determining how much an investment will grow over time. It represents the percentage of the principal that is paid as interest over one year. For example, in our exercise, the annual interest rate is 4%, which means every year, 4% of the initial investment is added as interest. Converting percentages to decimals is essential in financial calculations. To do this, simply divide the percentage by 100. So, an interest rate of 4% becomes 0.04 in the formula. This rate makes a significant difference in how the investment grows over time, especially when compounded annually. As seen in our example, an initial investment of \(100 or \)1000, when compounded annually at 4%, increases substantially over 20 years.
Investment Growth
Investment growth is a measure of how an initial amount of money increases over time due to factors like interest rates and compounding. The notion of compound interest plays a pivotal role here. Unlike simple interest, where interest is calculated only on the principal amount, compound interest means interest is calculated on both the initial amount and the accumulated interest. This phenomenon can cause investments to grow exponentially over time. Our exercise illustrates this growth by comparing the future values of \(100 and \)1000. Both amounts, when compounded annually at a 4% interest rate, show significant growth over 20 years. Specifically, \(100 grows to approximately \)219.11, while \(1000 grows to about \)2191.12. The interest earned on \(1000 after 20 years is approximately ten times that of \)100, demonstrating the powerful effect of compound interest on investment growth.

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Most popular questions from this chapter

Blood alcohol content (BAC) is the amount of alcohol present in your blood as you drink. It is calculated by determining how many grams of alcohol are present in 100 milliliters (1 deciliter) of blood. So if a person has 0.08 grams of alcohol per 100 milliliters of blood, the \(\mathrm{BAC}\) is \(0.08 \mathrm{~g} / \mathrm{dl} .\) After a person has stopped drinking, the BAC declines over time as his or her liver metabolizes the alcohol. Metabolism proceeds at a steady rate and is impossible to speed up. For instance, an average ( \(150-1 \mathrm{~b}\) ) male metabolizes about 8 to 12 grams of alcohol an hour (the amount in one bottle of beer^{3} ). The behavioral effects of alcohol are closely related to the blood alcohol content. For example, if an average 150 -lb male drank two bottles of beer within an hour, he would have a BAC level of 0.05 and could suffer from euphoria, inhibition, loss of motor coordination, and overfriendliness. The same male after drinking four bottles of beer in an hour would be legally drunk with a BAC of \(0.10 .\) He would likely suffer from impaired motor function and decision making, drowsiness, and slurred speech. After drinking twelve beers in one hour, he will have attained the dosage for stupor (0.30) and possibly death (0.40) . The following table gives the BAC of an initially legally drunk person over time (assuming he doesn't drink any additional alcohol). $$ \begin{array}{lccccc} \hline \text { Time (hours) } & 0 & 1 & 2 & 3 & 4 \\ \text { BAC (g/dI) } & 0.100 & 0.067 & 0.045 & 0.030 & 0.020 \\ \hline \end{array} $$ a. Graph the data from the table (be sure to carefully label the axes). b. Justify the use of an exponential function to model the data. Then construct the function where \(B(t)\) is the \(\mathrm{BAC}\) for time \(t\) in hours. c. By what percentage does the BAC decrease every hour? d. What would be a reasonable domain for your function? What would be a reasonable range? e. Assuming the person drinks no more alcohol, when does the BAC reach \(0.005 \mathrm{~g} / \mathrm{dl}\) ?

On November \(25,2003,\) National Public Radio did a report on Under Armour, a sports clothing company, stating that their "profits have increased by \(1200 \%\) in the last 5 years." a. Let \(P(t)\) represent the profit of the company during every 5-year period, with \(A_{0}\) the initial amount. Write the exponential model for the company's profit. b. Assuming an initial profit of \(\$ 100,000,\) what would be the profit in year 5 ? Year \(10 ?\) c. Determine the \(a n n u a l\) growth rate for Under Armour.

Represents an exponential function. Construct that function and then identify the corresponding growth or decay rate in percentage form. $$ \begin{array}{ccccc} \hline x & 0 & 1 & 2 & 3 \\ y & 225.00 & 221.63 & 218.30 & 215.03 \\ \hline \end{array} $$

(Graphing program required.) Graph the functions \(f(x)=30+5 x\) and \(g(x)=3(1.6)^{x}\) on the same grid. Supply the symbol \(<\) or \(>\) in the blank that would make the statement true. a. \(f(0)\)_____\(g(0)\) b. \(f(6)\)_____\(g(6)\) c. \(f(7)\)_____\(-g(7)\) d. \(f(-5)\)____\(g(-5)\) e. \(f(-6)\)_____\(-g(-6)\) f. As \(x \rightarrow+\infty\), \(f(x)\)_____\(g(x)\) g. As \(x \rightarrow-\infty\), \(f(x)\)_____\(g(x)\)

The per-capita consumption of bottled water was 8 gallons in 1990 and has been increasing yearly by a factor of \(1.088 .\) What was the per capita consumption of bottled water 12 years later?
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