Taxi Fare Calculation: How Many Kilometers Were Traveled?
Hey guys! Let's dive into a common math problem we might encounter in real life: calculating taxi fares. Imagine you hop into a cab, and there's a base charge plus a per-kilometer fee. How do you figure out the total distance you traveled? We're going to break down an example step-by-step, so you'll be a pro at this in no time. This article will delve into the mathematics of taxi fares, specifically addressing the question: If a taxi charges a base fare of $20 plus $12 per kilometer, and the total cost of a ride is $80, how many kilometers were traveled? Understanding this problem requires a grasp of basic algebraic principles and the ability to translate word problems into mathematical equations. We'll explore how to set up the equation, solve for the unknown variable (kilometers traveled), and interpret the result in the context of the problem. Whether you're a student looking to brush up on your math skills or simply curious about how to calculate taxi fares, this guide will provide a clear and concise explanation.
Understanding the Problem
So, let's understand the problem first. We know the taxi has a base fare – that's the initial charge you pay no matter how far you go. Then, there's an additional charge for each kilometer you travel. We also know the total cost of the ride. Our mission? To figure out how many kilometers were covered. Let's break it down, guys, to really grasp what we're dealing with. The problem states that there is a fixed base fare, which is a constant amount added to the total cost regardless of the distance traveled. This is a crucial piece of information because it forms the foundation of our equation. Additionally, there's a variable cost component: a charge per kilometer. This means the farther you travel, the more you'll pay, and this cost is directly proportional to the distance. The problem also provides the total cost of the ride, which is the sum of the base fare and the cost per kilometer multiplied by the number of kilometers traveled. By identifying these key components, we can begin to formulate a mathematical representation of the situation. We need to represent these components with variables and constants to build an equation that will allow us to solve for the unknown: the number of kilometers traveled. This initial understanding is paramount to setting up the problem correctly and ensuring an accurate solution.
Identifying the Key Information
Alright, let's put on our detective hats and identify the key information. In this scenario, the base fare is $20, the cost per kilometer is $12, and the total cost is $80. These are our golden nuggets of information that we'll use to crack the case! Breaking down the given information into distinct components is essential for translating a word problem into a mathematical equation. The base fare represents the initial cost incurred regardless of the distance traveled. It's a fixed amount that's always added to the final bill. The cost per kilometer is the variable component, directly proportional to the distance covered. This value represents the rate at which the fare increases for each kilometer traveled. Finally, the total cost is the sum of the base fare and the total cost for the distance traveled. Identifying these values allows us to represent the problem mathematically and determine the unknown variable, which in this case is the number of kilometers traveled. Without a clear understanding of these key components, it would be difficult to formulate the correct equation and solve for the desired result. Therefore, careful extraction and labeling of the given information are crucial steps in the problem-solving process.
Setting Up the Equation
Now comes the fun part – setting up the equation! We can express the total cost as the base fare plus the cost per kilometer multiplied by the number of kilometers. If we let 'x' represent the number of kilometers, our equation looks like this: $80 = $20 + ($12 * x). Let’s walk through the process of translating the word problem into a mathematical equation, step by step. The first step is to define a variable to represent the unknown quantity we're trying to find. In this case, we'll use 'x' to represent the number of kilometers traveled. Next, we need to express the total cost of the ride in terms of the given information and the variable 'x'. The total cost is the sum of the base fare and the cost per kilometer multiplied by the number of kilometers traveled. Mathematically, this can be expressed as: Total Cost = Base Fare + (Cost per Kilometer * Number of Kilometers). Now, we can substitute the given values into this equation. We know the total cost is $80, the base fare is $20, and the cost per kilometer is $12. Substituting these values, we get: $80 = $20 + ($12 * x). This equation represents the relationship between the total cost, the base fare, the cost per kilometer, and the number of kilometers traveled. It's a linear equation in one variable, which means we can solve for 'x' using basic algebraic techniques. Setting up the equation correctly is the most crucial step in solving the problem. It ensures that we accurately represent the relationships between the given quantities and the unknown variable. A well-formulated equation will lead us to the correct solution, while an incorrect equation will yield an inaccurate result.
Defining the Variable
Okay, so we need a superhero for our equation, and that superhero is our variable! Defining the variable is like giving our unknown a name. We're using 'x' to stand for the number of kilometers traveled. This makes it easier to keep track of what we're trying to find. When setting up an equation to solve a word problem, the first critical step is to define the variable that represents the unknown quantity we are trying to determine. This variable acts as a placeholder for the value we're looking for and allows us to express the relationships between the given information and the unknown in mathematical terms. In this particular problem, the unknown quantity is the number of kilometers traveled. Therefore, we define 'x' to represent this quantity. This choice is arbitrary; we could have used any other letter or symbol, but 'x' is a common convention in algebra. By clearly defining the variable, we establish a clear and concise representation of what we're trying to find. This clarity is essential for the subsequent steps in the problem-solving process, including formulating the equation and solving for the variable. Without a well-defined variable, it becomes difficult to translate the word problem into a mathematical equation and to interpret the solution in the context of the problem. Therefore, careful consideration should be given to defining the variable in a way that accurately represents the unknown quantity and facilitates the problem-solving process.
Solving the Equation
Time to put on our math hats and solve the equation! Our equation is $80 = $20 + ($12 * x). First, let's subtract $20 from both sides to isolate the term with 'x'. This gives us $60 = $12 * x. Now, we divide both sides by $12 to solve for 'x'. So, x = 5. Woohoo! Let's break down the step-by-step process of solving the equation $80 = $20 + ($12 * x) to determine the value of 'x', which represents the number of kilometers traveled. The first step in solving for 'x' is to isolate the term containing 'x' on one side of the equation. To do this, we need to eliminate the constant term, which is $20, from the right side of the equation. We can achieve this by subtracting $20 from both sides of the equation. This maintains the equality of the equation while simplifying it. Subtracting $20 from both sides, we get: $80 - $20 = $20 + ($12 * x) - $20, which simplifies to $60 = $12 * x. Now that we have isolated the term containing 'x', we need to solve for 'x' by eliminating the coefficient, which is $12. To do this, we divide both sides of the equation by $12. This again maintains the equality of the equation while isolating 'x'. Dividing both sides by $12, we get: $60 / $12 = ($12 * x) / $12, which simplifies to 5 = x. Therefore, the solution to the equation is x = 5. This means that the number of kilometers traveled is 5. The process of solving an equation involves performing operations on both sides of the equation to isolate the variable and determine its value. Each step in the process must maintain the equality of the equation to ensure that the solution remains valid. By following these steps carefully, we can solve for the unknown variable and answer the question posed in the problem.
Isolating the Variable
Alright, imagine 'x' is a shy little guy who wants to be alone. Our job is isolating the variable, meaning we want to get 'x' by itself on one side of the equation. We do this by performing the same operations on both sides of the equation to keep things balanced. When solving an algebraic equation, the primary goal is to isolate the variable on one side of the equation. This means manipulating the equation using mathematical operations until the variable is by itself on one side, with a constant value on the other side. Isolating the variable allows us to determine its value, which is the solution to the equation. The process of isolating the variable involves performing inverse operations on both sides of the equation to undo any operations that are being applied to the variable. For example, if the variable is being added to a constant, we subtract that constant from both sides of the equation. If the variable is being multiplied by a coefficient, we divide both sides of the equation by that coefficient. It is crucial to perform the same operation on both sides of the equation to maintain the equality of the equation. Any operation performed on one side must also be performed on the other side to ensure that the equation remains balanced and the solution remains valid. Isolating the variable is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. By mastering this technique, we can effectively solve equations and find the values of unknown variables.
Interpreting the Result
We've got our answer, but what does it mean? Interpreting the result is super important! We found that x = 5, which means the taxi traveled 5 kilometers. It's like translating math back into the real world. Now, guys, we have to make sure that the solution makes sense in the context of the original problem. Once we have obtained a solution to a mathematical problem, it is crucial to interpret the result in the context of the original problem. This involves understanding what the solution represents and whether it makes sense given the information provided in the problem. In the case of this taxi fare problem, we found that x = 5, where 'x' represents the number of kilometers traveled. Therefore, the solution means that the taxi traveled 5 kilometers. To ensure that this solution is reasonable, we can check if it satisfies the conditions stated in the problem. The problem states that the taxi charges a base fare of $20 plus $12 per kilometer, and the total cost of the ride was $80. If the taxi traveled 5 kilometers, the cost for the distance would be $12 * 5 = $60. Adding the base fare of $20, the total cost would be $20 + $60 = $80, which matches the total cost given in the problem. This confirms that our solution is correct and makes sense in the context of the problem. Interpreting the result is an essential step in the problem-solving process because it ensures that we understand the meaning of the solution and that it is consistent with the given information. It also helps us to avoid errors and to communicate the solution effectively. Without proper interpretation, a solution may be meaningless or even misleading.
Checking the Solution
Before we do a victory dance, let's check the solution! Plug x = 5 back into our original equation: $80 = $20 + ($12 * 5). Does it hold true? $80 = $20 + $60. Yep, $80 = $80! Our answer is correct! The final step in solving a mathematical problem is to check the solution. This involves verifying that the solution obtained satisfies the original equation or conditions stated in the problem. Checking the solution is crucial because it helps to identify any errors that may have occurred during the problem-solving process. In the case of the taxi fare problem, we found that x = 5, which means the taxi traveled 5 kilometers. To check this solution, we substitute x = 5 back into the original equation: $80 = $20 + ($12 * x). Substituting x = 5, we get: $80 = $20 + ($12 * 5). Simplifying the equation, we have: $80 = $20 + $60. Further simplification yields: $80 = $80. Since the left side of the equation equals the right side, the solution x = 5 satisfies the equation. This confirms that our solution is correct. Checking the solution can also involve verifying that the solution makes sense in the context of the problem. In this case, we determined that traveling 5 kilometers would result in a total cost of $80, which aligns with the information given in the problem. If the solution did not satisfy the equation or make sense in the context of the problem, it would indicate that an error has occurred and that the problem-solving process needs to be reviewed. Therefore, checking the solution is an essential step in ensuring the accuracy and validity of the result.
Conclusion
So there you have it, guys! We successfully calculated the distance traveled in the taxi. By understanding the problem, setting up the equation, solving for 'x', and interpreting the result, we nailed it! Remember, math is like a puzzle, and each step brings you closer to the solution. In conclusion, solving mathematical problems, such as the taxi fare calculation we've explored, involves a systematic approach that encompasses understanding the problem, setting up an equation, solving for the unknown variable, interpreting the result, and checking the solution. Each step in this process is crucial for arriving at an accurate and meaningful answer. The first step, understanding the problem, involves carefully reading and analyzing the given information to identify the key components and relationships. This includes identifying the known values, the unknown variable, and the conditions or constraints of the problem. Once the problem is understood, the next step is to set up an equation that mathematically represents the relationships between the known values and the unknown variable. This equation serves as a model of the problem and allows us to use algebraic techniques to solve for the unknown. Solving the equation involves manipulating the equation using mathematical operations until the variable is isolated on one side, allowing us to determine its value. This step requires a solid understanding of algebraic principles and techniques. After obtaining a solution, it is essential to interpret the result in the context of the original problem. This involves understanding what the solution represents and whether it makes sense given the information provided in the problem. Finally, the solution should be checked to ensure that it satisfies the original equation and conditions stated in the problem. This step helps to identify any errors that may have occurred during the problem-solving process. By following this systematic approach, we can effectively solve a wide range of mathematical problems and gain a deeper understanding of the underlying concepts. Keep practicing, and you'll become math whizzes in no time!