Problem 47 - Algebra 1

Question

You are taking a trip on the highway in a car that gets a gas mileage of about 26 miles per gallon for highway driving. You start with a full tank of 12 gallons of gasoline. Find your rate of gas consumption (the gallons of gas used to drive 1 mile).

Step-by-Step Solution

Answer

The rate of gas consumption is \(\frac{1}{26}\) gallons per mile.

Step 1: Understand the Relationship

The rate of gas consumption is the reciprocal of the gas mileage. That is, \[ Rate\_of\_gas\_consumption = \frac{1}{Gas\_mileage} \]

Step 2: Substitute Given Values

Substitute the given value of gas mileage 26 miles per gallon into the equation. \[ Rate\_of\_gas\_consumption = \frac{1}{26} \]

Step 3: Calculation

Calculate the value of the rate of gas consumption. Resulting value represents the gallons of gas used to drive 1 mile.

Key Concepts

Gas Mileage

Gas mileage, often referred to as fuel efficiency, measures how many miles a vehicle can travel per gallon of fuel. This figure is an essential aspect of understanding a car's performance, especially when planning for long trips or managing a budget.

For our scenario, having a gas mileage of 26 miles per gallon means that for each gallon of gas, the car can cover 26 miles. Knowing this number helps in estimating how far you can go with a certain amount of fuel and how often you would need to refill. In essence, the higher the gas mileage, the fewer pitstops and the less money spent on fuel.

Exercise Improvement Advice

When tackling problems related to gas mileage, it's crucial to understand the unit 'miles per gallon'. It's an indication of efficiency and helps determine your car's performance under different driving conditions, such as highway versus city. For a more in-depth understanding, students could compare the gas mileage of different vehicles or analyze how various driving habits might affect this number.

Reciprocal Relationship in Math

In mathematics, the concept of a reciprocal relationship involves two quantities that are inversely related to each other. To put it simply, if you have a number 'a', its reciprocal is '1/a'. The product of a number and its reciprocal is always 1. This is foundational in algebra and fractions.

The reciprocal relationship plays a vital role in various mathematical calculations and in understanding proportions. It also turns up in our real-world problem about gas mileage: The rate of gas consumption, which is how much fuel you use for one mile, is the reciprocal of gas mileage.

Connecting the Concepts

Understanding reciprocal relationships is particularly helpful when inverting proportions or rates. For example, if a car's gas mileage is 26 miles per gallon, its rate of gas consumption is the reciprocal, which means 1/26 gallons per mile. This is how we derive the rate at which the car uses fuel, a useful figure for various practical purposes like planning the refueling intervals on a trip.

Algebraic Problem-Solving

Algebraic problem-solving is a critical skill set in math that underscores finding unknown values through equations and formulas. It often involves recognizing patterns, understanding relationships between variables, and applying algebraic operations to solve for these unknowns.

In the context of our exercise, algebraic problem-solving comes into play when we invert the known gas mileage to find the rate of gas consumption. We start with the relationship between the rate and mileage, express this relationship algebraically, and manipulate the equation to solve for the desired value.

Step-by-Step Approach

To improve proficiency in algebraic problem-solving, one should practice setting up equations based on word problems, identifying the reciprocal relationships, and substituting the known values to find the unknowns. Regular practice using different types of problems can enhance a student's ability to navigate complex algebraic challenges effectively.