Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Hair Products A hair product company sells three types of hair products for $$\$ 30$$, $$\$ 20$$, and $$\$ 10$$ per unit. In one year, the total revenue for the three products was $$\$ 800,000$$, which corresponded to the sale of 40,000 units. The company sold half as many units of the $$\$ 30$$ product as units of the $$\$ 20$$ product. How many units of each product were sold?

Step-by-Step Solution

Answer

10,000 units of the $30 product, 20,000 units of the $20 product, and 10,000 units of the $10 product were sold.

Step 1: Define the Variables

Let \( x \) represent the number of units sold of the $30 product, \( y \) represent the number of units sold of the $20 product, and \( z \) represent the number of units sold of the $10 product.

Step 2: Form The Equations

From the problem three equations can be formed:\[1.\) 30x + 20y + 10z = 800,000 \text{ (representing the total revenue)}\] \[2.\) x + y + z = 40,000 \text{ (representing the total number of units sold)}\] \[3.\) x = 0.5y \text{ (representing the condition that half as many units of the $30 product were sold)}\]

Step 3: Substitute Equation 3 Into Equations 1 and 2

By substituting equation 3 into equation 1 and equation 2, one gets:\[1.\) 30(0.5y) + 20y + 10z = 800,000\] \[2.\) 0.5y + y + z = 40,000\] which simplify to:\[1.\) 15y + 20y + 10z = 800,000 \rightarrow 35y + 10z = 800,000\] \[2.\) 1.5y + z = 40,000\]

Step 4: Solve The System Of Equations

By multiplying equation 2 by 10, one gets:\[15y + 10z = 400,000\] Substracting this from the rearranged equation 1:\[35y + 10z - (15y + 10z) = 800,000 - 400,000 \] \[20y = 400,000\] Solving for \( y \) gives \( y = 20,000 \) units of the $20 product were sold. Substituting \( y = 20,000 \) into equation 3, one gets \( x = 0.5 \times 20,000 = 10,000 \) units of the $30 product were sold. Substituting \( y = 20,000 \) and \( x = 10,000 \) into equation 2, one gets \( 10,000 + 20,000 + z = 40,000 \) which solve for \( z \) gives \( z = 10,000 \) units of the $10 product were sold.

Key Concepts

Variable Definition

In algebra, a variable is a symbol used to represent an unknown value. It's a foundational tool for formulating and solving equations. In the context of word problems, variables are assigned to quantities that we are trying to find.

For example, in our hair product scenario, the variables x, y, and z were defined to represent the unknown quantities: the units of products sold at \(30, \)20, and $10 respectively. By defining these variables at the outset, we clarify our aim and give structure to the problem-solving process.

Forming Equations

Once variables are defined, we can then form equations based on the relationships and conditions given in the problem. This step translates the word problem into an algebraic representation, allowing us to use mathematical methods to find a solution.

In our exercise, the relationships given were the total revenue and the total units sold, as well as the relationship between the amounts sold of each product. These relationships were translated into three distinct equations which connect the variables in a system that describes the entire scenario.

Substitution Method

The substitution method is a powerful tool in solving systems of equations. It involves expressing one variable in terms of another and then substituting that expression into another equation. This method simplifies the system and can be particularly useful when one equation in a system is already solved for a variable.

In the given problem, we used substitution to replace x with 0.5y based on the given condition, reducing the number of variables and thus simplifying the equations.

Systematic Problem-Solving

Systematic problem-solving is a structured approach to tackling problems. It involves breaking down the problem into manageable steps and tackling each step one at a time. Following systematic problem-solving, we first identified what we needed to find, then established relationships through equations, substituted to reduce complexity, and finally solved the simplified system.

This methodical approach not only helps in solving complex problems but also in thoroughly understanding the process, reducing the chance of errors and oversights.

Algebraic Representation

An algebraic representation is an expression of a word problem in the form of algebraic equations. This means translating the context and language of the problem into mathematical symbols and operations.

In our hair products example, the algebraic representation was crucial in converting the business scenario into a solvable mathematical model, using variables and equations. This representation made it possible to apply algebraic methods such as substitution and simplification to arrive at the correct quantities of each product sold.