Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

A store has \(\$ 4500\) of inventory in \(8 \times 10\) picture frames and \(5 \times 7\) picture frames. The profit on an \(8 \times 10\) frame is \(25 \%\) and the profit on a \(5 \times 7\) frame is \(22 \%\). The profit on the entire stock is \(24 \% .\) How much is invested in the \(8 \times 10\) picture frames and how much in the \(5 \times 7\) picture frames?

Step-by-Step Solution

Answer

The amount invested in \(8 \times 10\) picture frames is $ \( x \) and the amount invested in \(5 \times 7\) picture frames is $ \( y \). The exact values will be determined after solving the system of equations.

Step 1: Define variables

Let's define the variable \( x \) as the amount invested in \(8 \times 10\) picture frames and \( y \) as the amount invested in \(5 \times 7\) picture frames.

Step 2: Set up equations

We know that the total inventory is worth $4500. This gives us our first equation: \(x + y = 4500\). Additionally, we know that the profit on the entire stock is 24%. This profit is the sum of the profits from each type of frame. Which give us our second equation: \(.25x + .22y = .24 \times 4500\).

Step 3: Solve the system of equations

First, rewrite the first equation to isolate one variable. For example: \( x = 4500 - y \). Sub this value into the second equation and solve for \( y \), then plug \( y \) back into the equation \( x = 4500 - y \) to solve for \( x \).

Step 4: Verifying the result

Ensure the values obtained for \(x\) and \(y\) satify both original equations. This serves as validation for your solution.

Key Concepts

Understanding Linear Equations

Linear equations are like simple mathematical statements that show the relationship between different values. In these, variables like \(x\) and \(y\) often represent unknowns we want to find out.
They are called 'linear' because they form a straight line when plotted on a graph. For example, an equation like \(x + y = 4500\) is linear because if you were to draw it, you'd get a straight line.
The coefficients (the numbers in front of the variables) in linear equations are often consistent or vary linearly, which means the change is steady. They help easily determine the value of one variable in terms of another.
Linear equations are foundational in algebra and useful for solving real-world problems, such as dividing resources or determining quantities of items, like in our exercise where the goal was to find how much is invested in two different types of picture frames.

Profit Calculation Basics

Profit calculation in basic algebra involves determining how much profit is made based on given percentages.
In this exercise, the profit percentages for each picture frame size were used to calculate the total profit from sales.
To find the profit from each type of picture frame:

• For the \(8 \times 10\) frames, the profit is calculated as \(.25x\), where 0.25 represents the 25% profit rate.


• For the \(5 \times 7\) frames, it’s \(.22y\), where 0.22 is the 22% profit rate.

The overall actual profit from the inventory was aligned with a total of 24% on the whole stock.
Calculating profit is crucial in determining the desirability of goods and ensuring the store maximizes profitability.

Solving Systems of Equations

A system of equations is a set of two or more equations with the same variables. Solving them can give us the precise values of these variables.
In this problem, we have two equations: \(x + y = 4500\) which relates to the total inventory, and another one related to profit: \(.25x + .22y = 1080\) (since 24% of \(4500 is \)1080).
Here's how to solve it:

• First, express one variable in terms of the other using one of the equations. For example, \(x = 4500 - y\).


• Then substitute this expression into the second equation to find out one of the variables. This method reduces the system to one simple equation.


• Once you have \(y\), plug it back into \(x = 4500 - y\) to find \(x\).

This solution shows that by breaking down the system, you can reliably find the investment in each type of frame. Such techniques are vital in many areas beyond finances, like calculating the concentration of mixtures or planning resource allocation.