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Chapter 3: Problem 12

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).

Short Answer

Expert verified
Investment at 10%: \(600, Investment at 20%: \)400

Step by step solution

01

- Define Variables

Let the amount invested at 10% be denoted as \( x \) and the amount invested at 20% be denoted as \( y \).
02

- Set Up Equations

From the problem, we have two pieces of information: the total investment which is \( x + y = 1000 \) and the total interest which is \( 0.10x + 0.20y = 140 \).
03

- Solve the System of Equations

First, solve for one variable in terms of the other. From \( x + y = 1000 \), we get \( y = 1000 - x \). Substitute this into the second equation: \( 0.10x + 0.20(1000 - x) = 140 \).
04

- Simplify the Equation

Expand and simplify the equation: \( 0.10x + 200 - 0.20x = 140 \), which simplifies to \( -0.10x + 200 = 140 \).
05

- Isolate x

Subtract 200 from both sides: \( -0.10x = -60 \). Divide by -0.10: \( x = 600 \).
06

- Solve for y

Using \( y = 1000 - x \), substitute \( x = 600 \) to get \( y = 1000 - 600 = 400 \).
07

- Verify the Solution

Check the amounts in the interest equation: \( 0.10(600) + 0.20(400) = 60 + 80 = 140 \), which matches the given total interest.

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Key Concepts

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

investment
In the context of this problem, we are dealing with two investments. The first step is to understand what an investment means. An investment is putting money into something with the hope of making more money. Here, the money is invested in high-technology companies and the goal is to earn interest. It鈥檚 important to identify the amounts being invested, which is what we do by defining our variables: let x be the amount invested at 10%, and y be the amount invested at 20%.
interest rates
Interest rates are crucial in determining how much money you earn on your investments. In this problem, we have two different interest rates: 10% and 20%. The interest is calculated by multiplying the amount invested by the interest rate. For instance, if you invest x dollars at 10%, the interest earned is 0.10x dollars. Similarly, if you invest y dollars at 20%, the interest earned is 0.20y dollars. Summing these gives us the total interest: 0.10x + 0.20y = 140.
algebraic solutions
To find the precise amounts of the investments, we need to solve a system of equations using algebra. We start by setting up our equations based on the problem statement:
  • The total investment:
    x + y = 1000
  • The total interest:
    0.10x + 0.20y = 140
Our goal is to find x and y using these equations. The next steps involve algebraic manipulation to solve these two equations, where we first solve one equation for one variable and then substitute that into the other equation.
substitution method
The substitution method is a powerful tool for solving systems of equations. Here鈥檚 how it works for our problem:
  • First, solve one equation for one variable. From the equation x + y = 1000, we solve for y: y = 1000 - x.
  • Next, substitute this expression into the second equation. This substitutes y in 0.10x + 0.20(1000 - x) = 140.
  • Expand and simplify: 0.10x + 200 - 0.20x = 140, simplifying to -0.10x + 200 = 140.
  • Solve for x: subtract 200 from both sides to get -0.10x = -60, and divide by -0.10, giving x = 600.
Lastly, we use y = 1000 - x to find y: y = 1000 - 600 = 400. Always verify your solution using both original equations to ensure accuracy.

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Most popular questions from this chapter

(Graphing program required.) A company manufactures a particular model of DVD player that sells to retailers for \(\$ 85\). It costs \(\$ 55\) to manufacture each DVD player, and the fixed manufacturing costs are \(\$ 326,000 .\) a. Create the revenue function \(R(x)\) for selling \(x\) number of DVD players. b. Create the cost function \(C(x)\) for manufacturing \(x\) DVD players. c. Plot the cost and revenue functions on the same graph. Estimate and interpret the breakeven point. d. Shade in the region where the company would make a profit. e. Shade in the region where the company would experience a loss. f. What is the inequality that represents the profit region?

(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?

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).

Explain what is meant by "a solution to a system of equations."

Construct a sketch of each system by hand and then estimate the solution(s) to the system (if any). $$ \begin{array}{ll} \text { a. } x+2 y=1 & \text { b. } x+y=9 \\ x+4 y=3 & 2 x-3 y=-2 \end{array} $$
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