Problem 49 - 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. Do the variables g and m vary directly, inversely, or neither? Explain.

Step-by-Step Solution

Answer

The variables g (gallons of gasoline) and m (miles the car can travel) neither vary directly nor inversely.

Step 1: Understand direct and inverse variation

In direct variation, as one quantity increases, the other quantity increases at the same rate. In inverse variation, as one quantity increases, the other quantity decreases.

Step 2: Apply definitions to the problem

In the given scenario, as the amount of gasoline decreases (because it is being used up with driving), the miles that can be travelled increases. Thus, as one (gallons) decreases, the other (miles) increases. Hence, this scenario does not represent direct variation.

Step 3: Consider inverse variation

In inverse variation, as one quantity increases, the other decreases. However, in our case, as the gasoline gallons decrease (g) from 12 to 0, the miles travelled (m) increase from 0 to a maximum distance (12 gallons x 26 miles/gallon = 312 miles). So, the quantity of gasoline and the miles travelled do not vary inversely either.

Step 4: Conclusion

Since the variables do not increase or decrease together (direct variation), nor does one variable increase as the other decreases (inverse variation), we can conclude that the variables g and m neither vary directly nor inversely.

Key Concepts

Direct Variation

Direct variation is an important algebraic concept where two variables are connected in such a way that if one variable changes, the other changes at a constant rate. Specifically, if we say that one variable varies directly as another, we mean there is a constant such that when you multiply this constant by the first variable, you get the second variable. Mathematically, it's represented as:
\[ y = kx \]
where \( y \) and \( x \) are the two variables and \( k \) is the non-zero constant of proportionality. In real-world terms, for example, if you work at a job where you are paid hourly, the amount of money you earn is directly proportional to the number of hours you work. The more hours you work, the more money you earn, following the formula:
\[ \text{Earnings} = \text{Hourly Rate} \times \text{Hours Worked} \].

Inverse Variation

In contrast to direct variation, inverse variation describes a scenario where one variable increases while the other decreases at a rate proportional to the inverse of the increase. This relationship is also known as an indirect or inverse proportion. The equation typically representing inverse variation is:
\[ y = \frac{k}{x} \]
where, as \( x \) increases, \( y \) decreases such that their product is a constant \( k \). A classic example of inverse variation is the relationship between speed and travel time. The faster you go (speed), the less time it takes to cover a certain distance. Hence, if you double your speed, your travel time for a fixed distance would be halved, assuming there are no other influencing factors.

Variables Relationship

Understanding the relationship between variables is critical in algebra and various fields involving data analysis. When examining two variables, it's essential to determine how they correlate: do they move together in the same direction (direct variation), do they move in opposite directions (inverse variation), or is there no discernible pattern between their movements (neither direct nor inverse variation)? In some cases, variables may have a more complex relationship that doesn't fit neatly into the categories of direct or inverse variation, requiring deeper analysis or more advanced mathematical models to describe accurately.

Algebraic Concepts

Algebraic concepts form the foundation of understanding relationships between variables and solving mathematical problems. They involve working with symbols, usually letters, to represent numbers in equations and functions. These concepts include operations on variables, factoring, exponents, and understanding different types of functions, including linear, polynomial, and rational functions. Grasping these algebraic notions equips students with the tools to solve for unknowns, analyze patterns, and predict outcomes, which are valuable skills both within and outside mathematically oriented disciplines.