Calculating Gasoline Costs Identifying Proportional Relationships
Hey guys! Today, we're diving into a super practical math problem that many of us face regularly: calculating the cost of gasoline. We'll break down how to understand and represent the proportional relationship between the total cost and the price per gallon. Imagine Prenell is at the gas station, filling up his car with regular unleaded gas that costs $1.89 per gallon. Our mission is to figure out which equations accurately show this relationship. So, let's put on our math hats and get started!
Before we jump into the equations, let's quickly recap what a proportional relationship means. In simple terms, two quantities are proportional if they increase or decrease at the same rate. This means that if you double one quantity, the other quantity also doubles. The key characteristic of a proportional relationship is that it can be represented by the equation y = kx, where:
yis one quantityxis the other quantitykis the constant of proportionality
This constant, k, is super important because it tells us the rate at which the two quantities change together. In our gas problem, the total cost (c) is proportional to the number of gallons (p) Prenell buys. The price per gallon ($1.89) is the constant of proportionality, as it's the fixed rate that connects the number of gallons and the total cost. Understanding this foundational concept is crucial because it allows us to model real-world situations with mathematical precision. For instance, if Prenell buys 2 gallons, the total cost will be 2 times $1.89. If he buys 10 gallons, it will be 10 times $1.89, and so on. This direct and consistent relationship is what defines proportionality, making it a fundamental concept in algebra and everyday problem-solving.
Okay, now that we've got a good grasp of proportional relationships, let's apply this to Prenell's gas situation. We need to figure out how to express the relationship between the total cost (c) and the number of gallons (p) when the price per gallon is $1.89. The key here is to identify which variable depends on the other. In this case, the total cost (c) depends on the number of gallons (p) Prenell purchases. This dependency is what drives the equation we'll construct. Think of it like this: the more gallons Prenell buys, the higher the total cost will be. So, the total cost is essentially a function of the number of gallons. Given that the price per gallon is the constant rate ($1.89), we can express this relationship using the proportional equation format we discussed earlier, y = kx. Here, y corresponds to the total cost (c), x corresponds to the number of gallons (p), and k is the constant price per gallon ($1.89). Substituting these values into our equation, we get c = 1.89p. This equation tells us that the total cost is equal to $1.89 multiplied by the number of gallons purchased. It's a direct representation of the proportional relationship at play.
But wait, there's another way to look at this! We can also rearrange this equation to solve for p in terms of c. To do this, we simply divide both sides of the equation c = 1.89p by 1.89. This gives us p = c / 1.89. This equation tells us the number of gallons Prenell can buy for a given total cost. Both equations represent the same proportional relationship, just from different perspectives. The first equation, c = 1.89p, is useful for calculating the total cost when we know the number of gallons. The second equation, p = c / 1.89, is useful for calculating how many gallons can be purchased for a specific amount of money.
Now that we've derived the equations, let's think about how we can recognize the correct ones in a multiple-choice scenario. Remember, the core concept here is the proportional relationship, which is always represented by a linear equation that passes through the origin (0,0). This means that if Prenell buys 0 gallons of gas, the total cost will be $0. This simple fact can help us eliminate any equations that don't fit this basic criterion. When looking at the options, keep an eye out for equations that match the forms we derived: c = 1.89p and p = c / 1.89. These equations are the direct mathematical translations of the proportional relationship between the total cost and the gallons of gas. Another tip is to test the equations with a simple scenario. For instance, if Prenell buys 1 gallon of gas, the total cost should be $1.89. Plug these values into the equations and see if they hold true. If an equation gives you a different result, it's likely incorrect. Always double-check the units as well. The total cost should be in dollars, and the number of gallons should be in gallons. Making sure the units are consistent will prevent errors.
Furthermore, be cautious of equations that might seem similar but don't accurately represent the proportional relationship. For example, an equation like c = p + 1.89 would be incorrect because it implies an additive relationship rather than a multiplicative one. This equation would suggest that the total cost is the number of gallons plus $1.89, which doesn't make sense in the context of buying gasoline. Similarly, equations that involve exponents or other non-linear operations are unlikely to represent a simple proportional relationship. By keeping these considerations in mind, you can confidently identify the correct equations that model Prenell's gasoline purchase.
The math we've been doing here isn't just theoretical; it's incredibly practical! Understanding proportional relationships helps us in all sorts of everyday situations, not just at the gas pump. Think about grocery shopping, for instance. The price of many items is proportional to their weight or quantity. If you know the price per pound of apples, you can easily calculate the cost of buying several pounds. This is the same principle we used for the gasoline problem. Similarly, if you're baking a cake and need to double the recipe, you're using proportional relationships to adjust the ingredients correctly. Each ingredient needs to be increased by the same factor to maintain the recipe's proportions.
Another common application is in currency exchange. The exchange rate between two currencies is a constant of proportionality. If you know the exchange rate between US dollars and Euros, you can calculate how many Euros you'll get for a certain amount of dollars, or vice versa. This is crucial for travelers and international business transactions. In the world of finance, interest rates often involve proportional relationships. The amount of interest you earn on a savings account is proportional to the principal amount and the interest rate. Understanding these relationships helps you make informed decisions about your finances. Even in more complex scenarios like calculating the distance traveled at a constant speed (distance = speed Ă— time), we're using the concept of proportionality. Recognizing these patterns empowers us to make accurate estimations and solve problems efficiently in numerous real-life contexts. So, the next time you're faced with a situation involving rates and quantities, remember the power of proportional relationships!
So, there you have it! We've taken a simple scenario – Prenell filling up his car with gas – and used it to explore the concept of proportional relationships. We've seen how to express these relationships mathematically, how to identify the correct equations, and how these principles apply in our daily lives. Remember, the key takeaway is that a proportional relationship is a consistent, multiplicative connection between two quantities, represented by the equation y = kx. Mastering this concept opens up a world of practical problem-solving skills. Whether you're calculating the cost of groceries, adjusting a recipe, or understanding financial rates, proportional relationships are a valuable tool in your mathematical toolkit. Keep practicing, and you'll become a pro at spotting and solving these problems! And remember, math isn't just about numbers; it's about understanding the world around us.
