Calculate Total Cost From Tip Percentage A Step-by-Step Solution
Have you ever encountered a situation where you know the tip amount and its fraction of the total bill, but you need to figure out the original cost of the meal? This is a common scenario, and understanding how to solve it is a valuable skill. In this comprehensive guide, we'll break down the problem-solving process, explore the underlying mathematical concepts, and illustrate the solution with a clear example. So, let's dive in and learn how to calculate the total cost of a meal when you know the tip amount and its fractional representation.
Understanding the Problem: Unveiling the Relationship Between Tip, Fraction, and Total Cost
Before we jump into the solution, it's crucial to understand the relationship between the tip amount, the fraction it represents, and the total cost of the food. In this scenario, the tip is a portion of the total cost, and the fraction expresses this portion mathematically. The core concept here is that the tip amount is equal to the fraction multiplied by the total cost. This relationship forms the basis of our equation and allows us to solve for the unknown total cost.
To illustrate this, consider a simple example. If the tip is $10 and it represents 1/5 (one-fifth) of the total cost, we know that $10 is 1/5 of the total cost. To find the total cost, we need to determine what number, when multiplied by 1/5, equals $10. This is the essence of the problem we're solving, and understanding this relationship is paramount to finding the correct solution. The key is to translate the word problem into a mathematical equation, which will then allow us to use algebraic techniques to isolate the unknown variable – in this case, the total cost of the meal.
We will translate this to . The total cost is the unknown, which we can represent with the variable x, leading to the equation . This equation is a concise mathematical representation of the problem statement. The fraction 3/20 multiplied by the total cost x equals the tip amount of $9. Solving this equation will reveal the value of x, which is the total cost of the food.
Setting Up the Equation: Translating Words into Math
The first step in solving this problem is to translate the given information into a mathematical equation. This involves identifying the key elements and their relationships. We are told that Tram left a tip of $9, which represents 3/20 of the total cost of the food. Let's represent the total cost of the food with the variable x. This is a crucial step in problem-solving – defining the unknown with a variable allows us to manipulate it mathematically.
The key phrase here is "3/20 of the total cost." In mathematics, the word "of" often indicates multiplication. Therefore, "3/20 of the total cost" can be written as (3/20) * x, or (3/20)x. We are also told that this amount is equal to $9. This gives us the equation: (3/20)x = 9. This equation is the heart of the solution. It accurately represents the relationship between the tip, the fraction, and the total cost.
This equation states that three-twentieths of the total cost x is equal to $9. Now, we have a clear mathematical statement that we can solve to find the value of x. The ability to translate word problems into mathematical equations is a fundamental skill in algebra and problem-solving. It allows us to use the power of mathematics to find solutions to real-world problems.
Solving for the Unknown: Isolating the Total Cost
Now that we have the equation (3/20)x = 9, we need to solve for x, which represents the total cost of the food. To isolate x, we need to undo the operation that is being performed on it. In this case, x is being multiplied by the fraction 3/20. The inverse operation of multiplication is division. However, dividing by a fraction can be tricky, so we use a more convenient method: multiplying by the reciprocal.
The reciprocal of a fraction is obtained by flipping the numerator and the denominator. The reciprocal of 3/20 is 20/3. The key principle here is that multiplying a fraction by its reciprocal equals 1. This is because (3/20) * (20/3) = 60/60 = 1. Multiplying both sides of the equation by the reciprocal will isolate x on one side.
So, we multiply both sides of the equation (3/20)x = 9 by 20/3. This gives us: (20/3) * (3/20)x = 9 * (20/3). On the left side, the (20/3) and (3/20) cancel each other out, leaving us with just x. On the right side, we have 9 * (20/3). We can simplify this by first dividing 9 by 3, which gives us 3, and then multiplying 3 by 20, which gives us 60. Therefore, the equation simplifies to x = 60. This means the total cost of the food was $60.
Verifying the Solution: Ensuring Accuracy and Understanding
Once we have a solution, it's crucial to verify its accuracy. This step ensures that we haven't made any calculation errors and that our answer makes sense in the context of the problem. To verify our solution, we can substitute the value of x back into the original equation and see if it holds true.
Our solution is x = $60. The original equation is (3/20)x = 9. Substituting x with 60, we get: (3/20) * 60 = 9. Now, we need to evaluate the left side of the equation. (3/20) * 60 can be simplified by first dividing 60 by 20, which gives us 3, and then multiplying 3 by 3, which gives us 9. So, the left side of the equation becomes 9, which is equal to the right side of the equation.
Since the equation holds true when we substitute x with $60, we can confidently say that our solution is correct. The total cost of the food was indeed $60. This verification step not only confirms our answer but also reinforces our understanding of the problem and the solution process. It's a valuable practice to adopt in any problem-solving scenario.
The Answer: Total Cost Revealed
After meticulously setting up the equation, solving for the unknown, and verifying our solution, we have arrived at the answer. The total cost of the food, represented by the variable x, is $60. Therefore, the correct answer is (B) 3/20 * x = 9; x = $60. This option accurately reflects the equation we derived from the problem statement and the value we calculated for the total cost.
This problem demonstrates the power of translating word problems into mathematical equations. By understanding the relationships between the given information and the unknown, we can set up an equation that allows us to solve for the desired value. The key steps involved in this process are: understanding the problem, setting up the equation, solving for the unknown, and verifying the solution. By following these steps, we can confidently tackle similar problems and develop our problem-solving skills.
Key Takeaways and Practical Applications: Beyond the Problem
This problem, while seemingly simple, highlights several key takeaways and has practical applications beyond just solving mathematical word problems. Firstly, it emphasizes the importance of understanding the relationships between different quantities. In this case, understanding the relationship between the tip, the fraction of the total cost, and the total cost itself was crucial for setting up the correct equation.
Secondly, it reinforces the fundamental algebraic skill of solving for an unknown. The ability to isolate a variable in an equation is a cornerstone of algebra and is essential for solving a wide range of problems. The technique of multiplying by the reciprocal is a particularly useful tool for solving equations involving fractions.
Thirdly, the importance of verification cannot be overstated. Checking our solution by substituting it back into the original equation ensures accuracy and helps us catch any potential errors. This practice builds confidence in our solutions and reinforces our understanding of the problem.
In terms of practical applications, this type of problem-solving skill is useful in various real-life scenarios. For example, when calculating tips at restaurants, splitting bills with friends, or determining discounts and sales prices. Understanding how to work with fractions and percentages is essential for managing personal finances and making informed decisions. The skills learned in solving this problem can be applied to a wide range of situations, making it a valuable exercise in mathematical reasoning and practical application.
In conclusion, by breaking down the problem into manageable steps, understanding the underlying concepts, and applying algebraic techniques, we successfully calculated the total cost of the food. This process not only provides the answer but also enhances our problem-solving skills and equips us to tackle similar challenges with confidence.
