Problem 70 - Elementary Algebra Within Reach

Question

A consumer buys \(g\) gallons of gasoline for a total of \(d\) dollars. Write an algebraic expression that represents the price per gallon.

Step-by-Step Solution

Answer

The algebraic expression for the price per gallon is \[ \frac{d}{g} \]

Step 1: Identify the Required Variables

In this problem, two variables are given. 'g' represents the gallons of gasoline bought by the consumer and 'd' represents the total cost of these gallons in dollars.

Step 2: Formulation of the Equation

The price per gallon can be calculated by dividing the total cost by the total number of gallons. Therefore, the algebraic expression that represents the price per gallon is \[ \frac{d}{g} \] where 'd' is the total cost and 'g' is the number of gallons.

Key Concepts

Understanding Price Calculation

Price calculation is a fundamental part of everyday life, especially when purchasing goods like gasoline. When calculating the price per unit, like per gallon in this case, you'll need to understand that it involves dividing the total cost by the total number of units (gallons).
This method helps break down larger quantities into smaller, more manageable pieces.By using the formula \[ \text{Price per gallon} = \frac{d}{g} \]you can easily find out how much each gallon costs. Here, \(d\) is the total amount in dollars, and \(g\) stands for the gallons purchased.
Practically speaking, this allows consumers to assess if they are getting a fair deal. If you know the price per gallon, comparing costs between different sellers becomes straightforward.

Effective Problem-Solving Approach

Tackling mathematical problems, like finding the price per gallon, with a structured approach is essential for success. Let’s break it down:

• Identify the Variables: Start by spotting the important variables given in the problem. Here, they are \(g\) and \(d\).


• Understand the Relationship: Know how these variables relate to each other. The total cost \(d\) divided by the total gallons \(g\) gives the price per gallon.


• Construct an Equation: Formulate your findings into an equation or expression. In this instance, it is \(\frac{d}{g}\).


• Solve: Use your equation to find the answer. This might mean substituting known values or simplifying the expression as needed.

By following these steps, you can systematically approach similar problems, enhancing your problem-solving skills with clarity and focus.
This structured methodology not only applies to math problems but can be tuned for real-life scenarios as well.

The Role of Variables in Algebra

Algebra heavily relies on variables to generalize mathematical expressions and equations. Variables are symbolic placeholders used to represent numbers whose values may change. In this context, we have been introduced to two primary variables: \(g\) and \(d\).

• \(g\) - Gallons of Gasoline: This variable represents the quantity of gasoline purchased and may vary depending on consumer needs.


• \(d\) - Total Dollars Spent: This signifies the total expenditure on gasoline and can fluctuate based on market prices and the quantity purchased.

By treating \(g\) and \(d\) as variables, algebra allows for an expression like \(\frac{d}{g}\) to be flexible and applicable across a range of situations.
Understanding and using variables effectively is a cornerstone of mastering algebra, enabling the translation of real-world situations into manageable algebraic terms.