Solving 12/N = 4/5 A Step-by-Step Guide To Finding N
In this comprehensive guide, we will walk you through the process of solving the equation 12/N = 4/5. This type of equation, known as a proportion, is a fundamental concept in mathematics and appears in various real-world applications. Mastering the techniques to solve these equations is crucial for building a strong foundation in algebra and problem-solving. We will break down each step, ensuring clarity and understanding, so that you can confidently tackle similar problems in the future. This guide aims to not only provide the solution but also to explain the underlying principles and reasoning, making it a valuable resource for students, educators, and anyone looking to enhance their mathematical skills. By the end of this article, you will be equipped with the knowledge and confidence to solve proportions effectively.
Understanding Proportions
Before diving into the solution, let's first understand what a proportion is. In mathematical terms, a proportion is an equation that states that two ratios or fractions are equal. The equation 12/N = 4/5 is a perfect example of a proportion. Here, 12/N represents one ratio, and 4/5 represents another. The 'N' in the equation is the unknown variable that we need to find. Solving a proportion means finding the value of this unknown variable that makes the equation true. Proportions are widely used in various fields, such as scaling recipes, converting units, and calculating percentages. They are a powerful tool for solving problems involving relationships between quantities. Understanding proportions is not only essential for solving algebraic equations but also for applying mathematical concepts in everyday situations. For instance, if you need to scale a recipe that serves 4 people to serve 10, you would use proportions to calculate the adjusted quantities of each ingredient. Similarly, in unit conversions, such as converting kilometers to miles, proportions are used to maintain the correct relationship between the units. Grasping the concept of proportions lays the groundwork for more advanced mathematical topics and enhances your ability to solve practical problems efficiently.
Step 1: Cross-Multiplication
The first step in solving the proportion 12/N = 4/5 is to use a technique called cross-multiplication. Cross-multiplication is a method used to simplify proportions and eliminate the fractions. It involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal to each other. In this case, we multiply 12 (the numerator of the first fraction) by 5 (the denominator of the second fraction), and we multiply 4 (the numerator of the second fraction) by N (the denominator of the first fraction). This process transforms the equation into a more manageable form without fractions. The cross-multiplication step is crucial because it converts the proportion into a linear equation, which is generally easier to solve. By eliminating the fractions, we can isolate the variable 'N' and determine its value. This step is not only applicable to this specific equation but is a universal technique for solving any proportion. It's a fundamental skill in algebra and is widely used in various mathematical contexts. Cross-multiplication provides a direct path to simplifying the equation and finding the unknown variable, making it an indispensable tool in your mathematical toolkit.
Applying cross-multiplication to our equation 12/N = 4/5, we get:
12 * 5 = 4 * N
This simplifies to:
60 = 4N
Step 2: Isolate the Variable N
After performing cross-multiplication, we are left with the equation 60 = 4N. The next step is to isolate the variable N. This means we need to get N by itself on one side of the equation. To do this, we need to undo the multiplication that is currently being applied to N. In this case, N is being multiplied by 4. The opposite operation of multiplication is division, so we will divide both sides of the equation by 4. Dividing both sides by the same number maintains the equality of the equation, ensuring that we are performing a valid algebraic manipulation. Isolating the variable is a fundamental concept in algebra, and it is used extensively in solving various types of equations. The goal is always to have the variable alone on one side, which allows us to directly read off its value. This step is crucial because it brings us closer to finding the solution of the equation. By dividing both sides by 4, we will effectively cancel out the multiplication on the side with N, leaving N isolated and revealing its value. This process demonstrates a key principle in algebra: performing inverse operations to solve for unknowns.
Dividing both sides of the equation 60 = 4N by 4, we get:
60 / 4 = (4N) / 4
This simplifies to:
15 = N
Step 3: Verify the Solution
After solving for N, it's always a good practice to verify the solution. This step ensures that the value we found for N is correct and that it satisfies the original equation. To verify our solution, we substitute the value we found for N back into the original equation and check if both sides of the equation are equal. This process helps us catch any potential errors we might have made during the solving process. Verification is a crucial step in problem-solving, not just in mathematics but in various other fields as well. It provides a check on our work and gives us confidence in our answer. By substituting the value back into the original equation, we are essentially reversing the steps we took to solve the equation and confirming that we arrive back at the starting point. This step is particularly important in more complex equations where the chances of making an error are higher. Verifying the solution not only confirms the correctness of the answer but also reinforces our understanding of the equation and the solution process. It's a valuable habit to develop as it promotes accuracy and attention to detail.
Substituting N = 15 back into the original equation 12/N = 4/5, we get:
12 / 15 = 4 / 5
To check if this is true, we can simplify the fraction 12/15 by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
(12 / 3) / (15 / 3) = 4 / 5
This simplifies to:
4 / 5 = 4 / 5
Since both sides of the equation are equal, our solution N = 15 is correct.
Final Answer
Therefore, the solution to the equation 12/N = 4/5 is:
N = 15
In conclusion, we have successfully solved the equation 12/N = 4/5 by following a step-by-step approach. We first understood the concept of proportions and how they represent the equality of two ratios. Then, we applied cross-multiplication to eliminate the fractions and simplify the equation. Next, we isolated the variable N by performing the inverse operation of division. Finally, we verified our solution by substituting the value of N back into the original equation to ensure its correctness. This process demonstrates a systematic method for solving proportions, a skill that is valuable in various mathematical contexts and real-world applications. Mastering these techniques will not only help you solve equations but also enhance your overall problem-solving abilities. The ability to break down complex problems into smaller, manageable steps is a crucial skill in mathematics and beyond. By understanding the underlying principles and practicing these steps, you can confidently tackle similar problems and build a strong foundation in algebra.
