Problem 8 - Intermediate Algebra

Question

A toy company makes a total of 500 puppets in three sizes during a production run. The small puppets cost \(\$ 5\) to make and sell for \(\$ 8\) each, the standard-size puppets cost \(\$ 10\) to make and sell for \(\$ 16\) each, and the super-size puppets cost \(\$ 15\) to make and sell for \(\$ 25 .\) The total cost to make the puppets is \(\$ 4,750\) and the revenue from their sale is \(\$ 7,700 .\) How many small, standard, and super-size puppets are made during a production run?

Step-by-Step Solution

Answer

There are 150 small, 250 standard, and 100 super-size puppets made.

Step 1: Define Variables

Let's define the variables to represent the number of each type of puppet produced: Let \( s \) represent the number of small puppets, \( n \) represent the number of standard-size puppets, and \( u \) represent the number of super-size puppets.

Step 2: Formulate the System of Equations

We have three equations based on the problem statement: 1. Total puppets: \( s + n + u = 500 \) 2. Total cost: \( 5s + 10n + 15u = 4750 \) 3. Total revenue: \( 8s + 16n + 25u = 7700 \)

Step 3: Simplify and Solve Equations

First, we can try to eliminate one of the variables. Start with the first two equations. Multiply the first equation by 5: \[ 5s + 5n + 5u = 2500 \]Subtract this from the second equation to eliminate \( s \):\[ (5s + 10n + 15u) - (5s + 5n + 5u) = 4750 - 2500 \]Simplify to get: \[ 5n + 10u = 2250 \]Divide the whole equation by 5: \[ n + 2u = 450 \]

Step 4: Substitute and Solve Further

Now use the simplified equation \( n + 2u = 450 \) with equation 1, \( s + n + u = 500 \). Substitute \( n = 450 - 2u \) into \( s + n + u = 500 \):\[ s + (450 - 2u) + u = 500 \]Simplify to find \( s \):\[ s + 450 - u = 500 \]\[ s = 50 + u \]

Step 5: Solve For Two Variables

Now substitute \( s = 50 + u \) and \( n = 450 - 2u \) into the third equation, the revenue equation:\[ 8(50 + u) + 16(450 - 2u) + 25u = 7700 \]Simplify:\[ 400 + 8u + 7200 - 32u + 25u = 7700 \]Combine like terms:\[ 7600 + u = 7700 \]Solve for \( u \): \[ u = 100 \]

Step 6: Find Remaining Values

Use \( u = 100 \) to find \( s \) and \( n \):\( s = 50 + u = 50 + 100 = 150 \)\( n = 450 - 2u = 450 - 200 = 250 \)

Key Concepts

Understanding Linear Equations

Linear equations are crucial in solving algebra problems that involve systems of equations. They are defined as equations where the highest power of the variable is one. In simpler terms, they look like lines when graphed on a coordinate axis. For example, the linear equation from the original problem, \( s + n + u = 500 \), represents a line in three-dimensional space. Similarly, the cost equation \( 5s + 10n + 15u = 4750 \) and the revenue equation \( 8s + 16n + 25u = 7700 \) are also linear.

Linear equations help us create relationships between variables, in this case, allowing us to model the relationships between the number of each puppet type and their associated costs and revenues. Solving a system of linear equations involves finding values for the variables that satisfy all equations simultaneously. Using simple techniques like substitution and elimination makes it possible to break down complex problems into simpler, manageable parts.

• They simplify complex relationships between variables.
• These equations are characterized by having constants and variables to the power of one.

Algebraic Problem-Solving Techniques

Algebraic problem-solving is a methodical process in which we use various strategies to find unknown values. The original problem was solved using techniques like substitution and elimination.

• Substitution: We solve one of the equations for one variable and plug it into another equation. For instance, we rearrange \( n + 2u = 450 \) to express \( n \) in terms of \( u \) and then substitute into other equations.
• Elimination: We eliminate one of the variables by combining equations. By subtracting one equation from another, variables are eliminated, making it simpler to solve. For example, \( 5s + 10n + 15u = 4750 \) was simplified to \( n + 2u = 450 \) by strategically subtracting another equation.

By using these techniques, we break down the problem into pieces, reducing complexity while finding accurate solutions.

Crafting a Mathematical Model

Mathematical modeling is the process of translating a real-world situation into mathematical language. By establishing equations that reflect the real scenario, we can then use math to find solutions.

In the puppet production problem, mathematical modeling is used to represent the number of small, standard, and super-sized puppets produced. Each of these quantities can be represented as variables in an equation that captures the total counts, total cost, and total revenue.

• We define variables to represent unknowns, such as \( s \) for small puppets.
• Equations are built based on the information provided, crafting a clear mathematical "picture" of the situation.
• We analyze the equations to draw conclusions and provide insightful answers to the original problem.

This approach not only aids in solving the current problem but is also a powerful tool for many other applications in real-world scenarios.