Question

Fuel Mixture Five hundred gallons of 89 -octane gasoline is obtained by mixing 87 -octane gasoline with 92 -octane gasoline. How much of each type of gasoline is required to obtain the specified mixture? (Octane ratings can be interpreted as percents.)

Short answer

The required amounts are 200 gallons of 87-octane gasoline and 300 gallons of 92-octane gasoline.

Step-by-step solution

Establish the first equation

From the exercise, it is known that the total quantity of gasoline is 500 gallons. If the amount of 87-octane gasoline is denoted as \(x\) (in gallons) and the amount of 92-octane gasoline as \(y\), then the equation is: \(x + y = 500\).

Establish the second equation

The exercise says that the mixture should be 89-octane, so the equation based on the octane rating of the mixture is: \(0.87x + 0.92y = 0.89 \times 500 = 445\).

Solve the system of equations

To solve the system, first solve the first equation for \(x\): \(x = 500 - y\). Substitute \(x\) from this equation into the second equation: \(0.87(500 - y) + 0.92y = 445\). After solving this equation for \(y\), get \(y = 300\). Substitute \(y = 300\) into the first equation to get \(x = 500 - 300 = 200\).

Key concepts

Octane Rating

Understanding octane rating is essential when tackling problems involving gasoline mixtures. Octane rating represents the quality of gasoline, indicating its ability to resist engine knocking or pinging during combustion. It's measured by the performance of the fuel under engine stress compared to a standard reference fuel. Higher octane ratings denote fuels that can withstand higher compression before detonating, which is vital in high-performance engines.

When an exercise mentions octane ratings, like in the provided example with 87-octane and 92-octane gasoline, it suggests that these can be treated as percentages that describe the proportion of high-octane components in the blend. In this context, if the aim is to create a mixture with an 89-octane rating, the student must find the right balance between the two gasolines to achieve the desired performance characteristics. Simplifying octane ratings to percentages allows for the creation of algebraic equations directly correlating to the volume of each type of gasoline needed.

Creating Algebraic Equations

Formulating algebraic equations is a fundamental skill for solving various mathematical and real-world problems. An equation is essentially a statement that two expressions are equal, often containing one or more unknowns, also called variables. When approaching an application problem like the fuel mixture exercise, the first step is translating the given information into such equations.

Begin by identifying the variables representing the amounts of different substances or items in the problem. Next, consider any given totals or overall requirements that provide a basis for an equation. In our example, the total quantity of the fuel mixture (500 gallons) and the desired octane rating (89) give us enough information to set up two different equations that relate the volumes of the individual components.

These equations must consider both the quantity and the octane quality, connecting the known values (total volume and desired octane rating) with the unknown values (amounts of each type of gasoline). By correctly creating these equations, we lay the groundwork for finding a solution through algebraic methods.

Solving Algebraic Problems

The process of solving algebraic problems involves finding the values of unknown variables that make the equations true. After creating the equations, the next steps usually include simplifying expressions, combining like terms, and employing methods like substitution or elimination to solve the system of equations.

In the fuel mixture problem, once the equations are set up based on the given information, we can substitute one equation into the other to express all terms in terms of a single variable. This maneuver reduces the complexity, transforming our original system of equations into a single equation that can be solved directly. The principles of balancing equations—whatever operation you do on one side, you must do on the other—ensure that we maintain equality throughout the solving process.

After isolating and finding the value of one variable, it's imperative to back-substitute that value into one of the original equations to find the value of the second variable. This step completes the solution and allows us to verify that both the quantity and octane rating requirements are satisfied, demonstrating not only math proficiency but also an understanding of the real-world implications of the problem and solution.