Problem 42
Question
A gas station sells three types of gas: Regular for \(\$ 3.00\) a gallon, Performance Plus for \(\$ 3.20\) a gallon, and Premium for \(\$ 3.30\) a gallon. On a particular day 6500 gallons of gas were sold for a total of \(\$ 20,050 .\) Three times as many gallons of Regular as Premium gas were sold. How many gallons of each type of gas were sold that day?
Step-by-Step Solution
Verified Answer
1500 gallons of Premium, 500 gallons of Performance Plus, 4500 gallons of Regular gas were sold.
1Step 1: Define Variables
Let's assign variables to the quantities we need to determine. Let \( x \) be the gallons of Premium gas sold, \( y \) be the gallons of Performance Plus gas sold, and \( z \) be the gallons of Regular gas sold.
2Step 2: Translate the Problem into Equations
Based on the problem description, we can write the following equations:\1. The total gallons equation: \( x + y + z = 6500 \) \2. The total sales equation: \( 3.30x + 3.20y + 3.00z = 20050 \) \3. The relationship between Regular and Premium gas: \( 3x = z \)
3Step 3: Substitute and Simplify
Use the relationship from equation 3 (\( z = 3x \)) to substitute \( z \) in the first and second equations:\- Substitute in total gallons equation: \( x + y + 3x = 6500 \), simplifying to \( 4x + y = 6500 \). \- Substitute in total sales equation: \( 3.30x + 3.20y + 3.00(3x) = 20050 \), simplifying to \( 3.30x + 3.20y + 9x = 20050 \), further simplifying to \( 12.30x + 3.20y = 20050 \).
4Step 4: Solve the System of Equations
With the simplified equations: \1. \( 4x + y = 6500 \) \2. \( 12.30x + 3.20y = 20050 \) \First, solve the first equation for \( y \): \\( y = 6500 - 4x \). \Substitute \( y \) in the second equation: \\( 12.30x + 3.20(6500 - 4x) = 20050 \). \Simplify the equation: \\( 12.30x + 20800 - 12.80x = 20050 \), \\( -0.50x = -750 \). \Solve for \( x \): \\( x = 1500 \) gallon of Premium gas.
5Step 5: Calculate Other Variables
Using \( x = 1500 \): \1. Calculate \( z \): \( z = 3x = 3 \times 1500 = 4500 \) gallons of Regular gas. \2. Calculate \( y \): Substitute \( x = 1500 \) into \( y = 6500 - 4x \), \\( y = 6500 - 4 \times 1500 = 500 \) gallons of Performance Plus gas.
Key Concepts
Variables and ExpressionsEquation SolvingMathematical Modeling
Variables and Expressions
In mathematics, particularly in solving word problems, we often use variables and expressions.
Variables are symbols that represent numbers. For instance, in our gas station problem, these variables help us represent unknown quantities of gas types sold.
### Using Variables * **Premium Gas (\(x\)**): This is the variable denoting the gallons of Premium gas sold.* **Performance Plus Gas (\(y\)**): Represents the gallons of Performance Plus gas sold.* **Regular Gas (\(z\)**): This variable signifies the gallons of Regular gas sold.### Building Expressions Expressions combine variables with constants to form equations. When translated from a real-world scenario, expressions help to model the problem statement effectively. For example:* **Gallons Expression**: The equation \(x + y + z = 6500\) is formed using our gas variables, summarizing the total gallons.* **Sales Expression**: It's the combination of prices with variables \(3.30x + 3.20y + 3.00z = 20050\) to reflect total sales.
By using variables and crafting expressions, mathematical problems are transformed into manageable equations.
Variables are symbols that represent numbers. For instance, in our gas station problem, these variables help us represent unknown quantities of gas types sold.
### Using Variables * **Premium Gas (\(x\)**): This is the variable denoting the gallons of Premium gas sold.* **Performance Plus Gas (\(y\)**): Represents the gallons of Performance Plus gas sold.* **Regular Gas (\(z\)**): This variable signifies the gallons of Regular gas sold.### Building Expressions Expressions combine variables with constants to form equations. When translated from a real-world scenario, expressions help to model the problem statement effectively. For example:* **Gallons Expression**: The equation \(x + y + z = 6500\) is formed using our gas variables, summarizing the total gallons.* **Sales Expression**: It's the combination of prices with variables \(3.30x + 3.20y + 3.00z = 20050\) to reflect total sales.
By using variables and crafting expressions, mathematical problems are transformed into manageable equations.
Equation Solving
Equation solving is the process of finding the values of variables that satisfy a set of equations. In our gas station problem, we're working with a system of equations. These equations arise from translating the problem into mathematical forms.
### Key Steps in Equation Solving 1. **Identify Known Values**: Start by understanding the equations based on the problem context. * Our known in this example: total gallons and total sales amount.2. **Substitution**: Utilize relationships between variables to reduce the number of variables. Here, the equation \(z = 3x\) allows substitution that simplifies other equations.3. **Simplification**: Break down equations by substituting the known variables, leading to simpler forms. * Example: \(4x + y = 6500\) and \(12.30x + 3.20y = 20050\) derived from initial translations.4. **Solve for Variables**: Use arithmetic to find specific variable values. * Finding \(x\) as 1500 gallons of Premium gas shows a successful solve.Equation solving requires systematic work through each step, maintaining the balance of each equation while isolating variables.
### Key Steps in Equation Solving 1. **Identify Known Values**: Start by understanding the equations based on the problem context. * Our known in this example: total gallons and total sales amount.2. **Substitution**: Utilize relationships between variables to reduce the number of variables. Here, the equation \(z = 3x\) allows substitution that simplifies other equations.3. **Simplification**: Break down equations by substituting the known variables, leading to simpler forms. * Example: \(4x + y = 6500\) and \(12.30x + 3.20y = 20050\) derived from initial translations.4. **Solve for Variables**: Use arithmetic to find specific variable values. * Finding \(x\) as 1500 gallons of Premium gas shows a successful solve.Equation solving requires systematic work through each step, maintaining the balance of each equation while isolating variables.
Mathematical Modeling
Mathematical modeling translates real-world scenarios into mathematical expressions to analyze and solve them. The gas station problem showcases this concept vividly.
### Translating Real-World Problems To model a situation mathematically, like sales at a gas station: * **Define Problem Conditions**: List out what is known - prices, sales amounts, and relationships. * **Establish Relationships**: Create equations that mirror real-life constraints and data. * Example: Regular gas being thrice the Premium gas. * Link quantities and prices through equations to reflect the scenario accurately. ### The Power of Modeling * **Visualization**: Modeling aids in visualizing and understanding complex scenarios. * **Predict Solutions**: Predict outcomes and verify them with actual data. * **Modify and Adapt**: It's flexible; you can adjust if conditions change or assumptions are off. In essence, mathematical modeling is a powerful tool that provides clarity and precision, turning intricate problems into solvable equations.
### Translating Real-World Problems To model a situation mathematically, like sales at a gas station: * **Define Problem Conditions**: List out what is known - prices, sales amounts, and relationships. * **Establish Relationships**: Create equations that mirror real-life constraints and data. * Example: Regular gas being thrice the Premium gas. * Link quantities and prices through equations to reflect the scenario accurately. ### The Power of Modeling * **Visualization**: Modeling aids in visualizing and understanding complex scenarios. * **Predict Solutions**: Predict outcomes and verify them with actual data. * **Modify and Adapt**: It's flexible; you can adjust if conditions change or assumptions are off. In essence, mathematical modeling is a powerful tool that provides clarity and precision, turning intricate problems into solvable equations.
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