Motorcycle Mileage: Calculate Distance Per Gallon

Hey guys! Let's dive into a super practical math problem today: figuring out how far a motorcycle can travel based on its gas mileage. We know this particular motorcycle can travel 48 miles on just one gallon of gas. That's pretty awesome, right? We're going to use this information, which is called the unit rate, to fill out a table and see how many miles the motorcycle can travel with different amounts of gas. So, buckle up (or should I say, helmet on?) and let's get started!

Understanding Unit Rate

First off, let's break down what a unit rate actually means. In simple terms, a unit rate tells us how much of something we get for one unit of something else. In our case, it's the number of miles the motorcycle travels for one gallon of gas. We already know this is 48 miles per gallon (mpg). This is our magic number, the key to unlocking all the other distances. Understanding the unit rate is crucial because it acts as our foundation for calculating distances for any number of gallons. It's like the single ingredient that makes the whole recipe work! Think of it this way: if you know how many cookies you can make with one cup of flour, you can easily figure out how many cookies you can make with two, three, or even ten cups of flour. The unit rate is your superpower in these kinds of calculations.

When we talk about the unit rate, we're essentially establishing a direct relationship between two quantities: miles traveled and gallons of gas used. This relationship is proportional, meaning that as one quantity increases, the other increases at a constant rate. In our scenario, every time we add a gallon of gas, the motorcycle travels an additional 48 miles. This consistent relationship allows us to use simple multiplication to find the distances for multiple gallons. So, mastering the concept of unit rate isn't just about solving this particular problem; it's about understanding a fundamental principle that applies to countless real-world situations, from calculating fuel efficiency to scaling recipes to determining the cost of goods.

Filling the Table: Miles Traveled vs. Gallons Used

Now, let's get to the fun part: filling out the table! We already have the first row: 48 miles for 1 gallon of gas. The table looks like this:

Our mission is to find the values for 'a', 'b', and 'c'. Since we know the motorcycle travels 48 miles per gallon, we can use simple multiplication to find the distances for 2, 3, and 4 gallons.

Calculating 'a': Miles Traveled with 2 Gallons

To find 'a', we need to figure out how many miles the motorcycle can travel with 2 gallons of gas. We know it travels 48 miles per gallon, so with 2 gallons, it will travel 48 miles/gallon * 2 gallons = 96 miles. So, a = 96. This calculation is pretty straightforward: we're essentially doubling the distance the motorcycle travels on one gallon of gas. Multiplication is our key tool here, allowing us to scale the distance proportionally to the amount of gas used. It's like saying,