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Chapter 1: Problem 85

Finance A total of 20,000 dollar is to be invested, some in bonds and some in certificates of deposit (CDs). If the amount invested in bonds is to exceed that in CDs by 3000 dollar, how much will be invested in each type of investment?

Short Answer

Expert verified
8500 dollars in CDs and 11500 dollars in bonds.

Step by step solution

01

Define Variables

Let the amount invested in CDs be denoted as \( x \) dollars. Then, the amount invested in bonds will be \( x + 3000 \) dollars.
02

Set Up Equation

The total amount invested is 20,000 dollars. Therefore, the equation can be formed as follows: \[ x + (x + 3000) = 20000 \]
03

Simplify the Equation

Combine like terms in the equation: \[ 2x + 3000 = 20000 \]
04

Solve for x

Isolate \( x \) by subtracting 3000 from both sides: \[ 2x = 20000 - 3000 \] \[ 2x = 17000 \] Then, divide by 2: \[ x = \frac{17000}{2} = 8500 \]
05

Determine the Amount in Bonds

Since the amount in bonds is \ x + 3000 \, calculate it as follows: \[ 8500 + 3000 = 11500 \]

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Key Concepts

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

linear equations
Linear equations are equations where the highest power of the variable is 1. They have the general form of \(ax + b = 0\), where \(a\) and \(b\) are constants. These equations are fundamental in algebra because they allow us to find unknown values. In our investment problem, the linear equation helps us balance the total investments between bonds and CDs. By setting up a linear equation, \(x + (x + 3000) = 20000\), we can find out how much is invested in each.
variable definition
In solving word problems, defining variables is crucial. This helps convert real-world scenarios into mathematical equations. In our problem, we define the amount invested in CDs as \(x\) dollars. Treating this unknown amount as \(x\) helps us write an equation. By saying the amount in bonds is \(x + 3000\), we acknowledge that the bond investment exceeds the CD investment by 3000 dollars. Accurate variable definition simplifies problem-solving.
solving equations
Solving equations involves finding the value of the variables that make the equation true. Once the linear equation \(x + (x + 3000) = 20000\) is set up, we can simplify it to combine like terms: \[2x + 3000 = 20000\]. To isolate \(x\), we subtract 3000 from both sides: \[2x = 17000\], and then divide by 2: \[x = 8500\]. This is an important skill as it is used for solving various real-world problems.
financial arithmetic
Financial arithmetic involves mathematical calculations related to finance. In our problem, the total investment is 20000 dollars. To ensure that this amount is distributed correctly between bonds and CDs, we use basic arithmetic operations like addition and subtraction. The final values found are: 8500 dollars in CDs and 11500 dollars in bonds. This type of arithmetic is crucial for effective financial planning and investments.

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Most popular questions from this chapter

On a recent flight from Phoenix to Kansas City, a distance of 919 nautical miles, the plane arrived 20 minutes early. On leaving the aircraft, I asked the captain, "What was our tail wind?" He replied,"I don't know, but our ground speed was 550 knots." Has enough information been provided for you to find the tail wind? If possible, find the tail wind. \((1 \mathrm{knot}=1\) nautical mile per hour)

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A civil engineer relates the thickness \(T,\) in inches, and height \(H,\) in feet, of a square wooden pillar to its crushing load \(L\), in tons, using the model \(T=\sqrt[4]{\frac{L H^{2}}{25}}\). If a square wooden pillar is 4 inches thick and 10 feet high, what is its crushing load?

Critical Thinking You are the manager of a clothing store and have just purchased 100 dress shirts for $20.00 each. After 1 month of selling the shirts at the regular price, you plan to have a sale giving 40% off the original selling price. However, you still want to make a profit of 4 on each shirt at the sale price. What should you price the shirts at initially to ensure this? If, instead of 40% off at the sale, you give 50% off, by how much is your profit reduced?

Challenge Problem Show that the real solutions of the equation \(a x^{2}+b x+c=0, a \neq 0,\) are the negatives of the real solutions of the equation \(a x^{2}-b x+c=0\). Assume that \(b^{2}-4 a c \geq 0\)
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