Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

A store has \(\$ 40,000\) of inventory in notebook computers and tablet computers. The profit on a notebook computer is \(20 \%\) and the profit on a tablet computer is \(25 \%\). The profit for the entire stock is \(24 \% .\) How much is invested in notebook computers and how much in tablet computers?

Step-by-Step Solution

Answer

The investment made in notebook computers and tablet computers are \$20,000 each.

Step 1: Define Variables

Let the investment on notebook computers be \(x\) dollars and the investment on tablet computers be \(y\) dollars.

Step 2: Formulate the Equations

From the problem, you have two key pieces of information which you can use to form a system of two linear equations. Firstly, the total investment is \$40,000, giving the equation \(x + y = 40000\). Secondly, the overall profit is 24%, with notebook computers offering a profit margin of 20% and tablet computers 25%. This information forms the second equation: \(0.20x + 0.25y = 0.24 * 40000\).

Step 3: Solve the System of Equations

You can use several methods to solve the system of equations, such as substitution, elimination or matrix method. Here, you'll use substitution. Isolate one variable from one equation (for example \(y = 40000 - x\)) and substitute this in the other equation. Solve for \(x\), then substitute \(x\) in the first equation to calculate \(y\).

Step 4: Calculation and Result

Substituting \(y = 40000 - x\) in the equation \(0.20x + 0.25*(40000 - x) = 9600\) gives \(x = 20000\) and \(y = 20000\). Thus, investment on both notebook computers and tablet computers is \$20,000 each.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. In the given problem, we have two variables: the investment in notebook computers, denoted as \(x\), and the investment in tablet computers, represented as \(y\). The problem statement provides us with two essential pieces of information, which we translate into two linear equations. The first equation indicates the total investment: \(x + y = 40,000\). The second equation comes from the profit calculation, relating the profit margins to the total profit: \(0.20x + 0.25y = 0.24 \times 40,000\). These equations form our system which we solve simultaneously to find the values of \(x\) and \(y\). Having a system of equations allows us to use different methods to find the values of variables that satisfy all conditions given by the problem.

Profit Calculation

Profit calculation in this problem involves knowing the percentage profit for each type of computer and the overall profit from all investments. The notebook computers provide a 20% profit, while tablet computers yield a 25% profit. We are told that the overall profit is 24% on the entire investment of \(\$40,000\).

To better understand:

• The profit from notebook computers would be \(0.20 \times x\).
• The profit from tablet computers would be \(0.25 \times y\).
• Thus, the total profit, which must match the overall profit, is expressed as \(0.20x + 0.25y = 0.24 \times 40,000\).

Understanding how to set up and work with this equation is pivotal to solving not just this problem but any problem involving profit calculations across investments with varying returns.

Investment Distribution

The distribution of investments in notebook and tablet computers is what we're looking to find out. Given that the total investment sums to \(\$40,000\), we assign \(x\) to the investment in notebook computers, and \(y\) for tablet computers.

The goal is to determine how this total investment is split between the two. When each investment proportion aligns with their respective profit margins to maintain an overall 24% profit, we can use the system of equations formed to solve for this distribution.

• Equation 1: Total investment, \(x + y = 40,000\).
• Equation 2: Profit alignment, \(0.20x + 0.25y = 0.24 \times 40,000\).

Ultimately, solving these equations will reveal the exact amount invested in each type of computer.

Substitution Method

The substitution method is one way to solve a system of equations. It involves solving one of the equations for one variable and substituting that expression into the other equation. Here's how it's done in our exercise:

1. Identify an equation where one variable can be easily isolated. We use \(x + y = 40,000\) to express \(y\) in terms of \(x\):

• \(y = 40,000 - x\)

2. Substitute this expression for \(y\) into the second equation:

• \(0.20x + 0.25(40,000 - x) = 9,600\)

3. Simplify and solve for \(x\):

• This results in \(x = 20,000\)

4. Use the value of \(x\) to find \(y\):

• Since \(y = 40,000 - x\), then \(y = 20,000\)

Through substitution, we easily determine that both investments are \(\$20,000\), making it a very effective technique for solving such problems.