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Chapter 3: Problem 11

Assume you have \(\$ 2000\) to invest for 1 year. You can make a safe investment that yields \(4 \%\) interest a year or a risky investment that yields \(8 \%\) a year. If you want to combine safe and risky investments to make \(\$ 100\) a year, how much of the \(\$ 2000\) should you invest at the \(4 \%\) interest? How much at the \(8 \%\) interest? (Hint: Set up a system of two equations in two variables, where one equation represents the total amount of money you have to invest and the other equation represents the total amount of money you want to make on your investments.)

Short Answer

Expert verified
Invest \$1500 at 4\text{\textpercent} interest and \$500 at 8\text{\textpercent} interest.

Step by step solution

01

Identify the Variables

Let \( x \) be the amount invested at \( 4\text{\textpercent} \) interest and \( y \) be the amount invested at \( 8\text{\textpercent} \) interest.
02

Set up the Total Investment Equation

The total amount invested is \$2000. Thus, the equation is: \[ x + y = 2000 \]
03

Set up the Total Interest Equation

The total interest earned should be \$100. The interest from \( x \) amount invested at \( 4\text{\textpercent} \) is \( 0.04x \) and from \( y \) amount invested at \( 8\text{\textpercent} \) is \( 0.08y \). Thus, the equation is: \[ 0.04x + 0.08y = 100 \]
04

Solve the System of Equations

Solve the system of equations by substitution or elimination method. Substitution method here: From the first equation, \( y = 2000 - x \). Substitute \( y \) in the second equation: \[ 0.04x + 0.08(2000 - x) = 100 \] Simplify and solve for \( x \): \[ 0.04x + 160 - 0.08x = 100 \] \[ -0.04x + 160 = 100 \] \[ -0.04x = -60 \] \[ x = 1500 \]
05

Calculate the Amount for the Risky Investment

Now that we have \( x = 1500 \), substitute \( x \) back into the first equation to find \( y \): \[ y = 2000 - 1500 = 500 \]
06

Conclusion

You should invest \$1500 at the \( 4\text{\textpercent} \) interest rate and \$500 at the \( 8\text{\textpercent} \) interest rate to earn \ $100.

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Key Concepts

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

System of Equations
An essential tool for solving investment problems is understanding 'system of equations'. It involves two or more equations with two or more variables. Here, we had two variables - the amounts invested at different interest rates. By setting up two distinct equations, each representing a different aspect of the problem, we can solve for unknowns.
For example, in this problem:
  • The first equation represented the total amount invested: \( x + y = 2000 \)
  • The second equation represented the total interest earned from investments: \( 0.04x + 0.08y = 100 \)
Creating and solving such systems helps decipher multiple unknowns systematically.
Simple Interest Calculations
Simple interest calculations are straightforward but crucial in investment problems. Interest is calculated annually and is dependent on the principal amount, rate of interest, and the time. The formula for simple interest is:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
In this scenario:
  • Interest earned from the safe investment at 4% is \( 0.04x \)
  • Interest earned from the risky investment at 8% is \( 0.08y \)
Both these contributions added to make the total interest of $100. Understanding this calculation is pivotal for breaking down combined interest scenarios in investments.
Substitution Method
The substitution method is a technique to solve systems of equations. It involves solving one equation for one variable and substituting this into the other equation. In this investment problem, we first had: \( x + y = 2000 \). Solving for \( y \), we got: \[ y = 2000 - x \]. Then this value of \( y \) was substituted into the second equation: \[ 0.04x + 0.08(2000 - x) = 100 \]
This simplifies the problem to one equation with one variable, enabling easier solving steps to find individual amounts invested. The substitution method is especially useful when one equation is simple and enables easy transposition.
Elimination Method
The elimination method, another way to solve systems of equations, involves eliminating one variable by adding or subtracting equations. While not used directly in this problem, it鈥檚 important to understand it as an alternative.
Explore this scenario:
  • From the equations \( x + y = 2000 \) and \( 0.04x + 0.08y = 100 \), you could multiply the first by 0.04 and get: \[ 0.04x + 0.04y = 80 \]
  • Subtracting this from the second equation: \[ 0.08y - 0.04y = 100 - 80 \]
This also results in a single equation with one variable, simplifying the solving process. Understanding both elimination and substitution methods offers flexibility and ease in solving a range of algebraic problems.

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Most popular questions from this chapter

If \(y=b+m x\), solve for values for \(m\) and \(b\) by constructing two linear equations in \(m\) and \(b\) for the given sets of ordered pairs. a. When \(x=2, y=-2\) and when \(x=-3, y=13\). b. When \(x=10, y=38\) and when \(x=1.5, y=-4.5\).

a. Solve the following system algebraically: $$ \begin{aligned} x+3 y &=6 \\ 5 x+3 y &=-6 \end{aligned} $$ b. Graph the system of equations in part (a) and estimate the solution to the system. Check your estimate with your answers in part (a).

(Graphing program recommended.) The blood alcohol concentration (BAC) limits for drivers vary from state to state, but for drivers under the age of 21 it is commonly set at 0.02. This level (depending upon weight and medication levels) may be exceeded after drinking only one 12 -oz can of beer. The formula $$ N=6.4+0.0625(W-100) $$ gives the number of ounces of beer, \(N,\) that will produce a BAC legal limit of 0.02 for an average person of weight \(W\). The formula works best for drivers weighing between 100 and \(200 \mathrm{lb}\). a. Write an inequality that describes the condition of too much blood alcohol for drivers under 21 to legally drive. b. Graph the corresponding equation and label the areas that represent legally safe to drive, and not legally safe to drive conditions. c. How many ounces of beer is it legally safe for a 100 -lb person to consume? A 150-lb person? A 200-lb person? d. Simplify your formula in part (b) to the standard \(y=m x+b\) form. e. Would you say that " 6 oz of beer +1 oz for every \(20 \mathrm{lb}\) over \(100 \mathrm{lb}\) " is a legally safe rule to follow?

Two investments in high-technology companies total \(\$ 1000 .\) If one investment earns \(10 \%\) annual interest and the other earns \(20 \%,\) find the amount of each investment if the total interest earned is \(\$ 140\) for the year (clearly in dot com days).

a. Solve the following system algebraically: $$ \begin{array}{l} S=20,000+2500 n \\ S=25,000+2000 n \end{array} $$ b. Graph the system in part (a) and use the graph to estimate the solution to the system. Check your estimate with your answer in part (a).
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