Evaluate M + 1 1/3 When M = 3 1/2: Step-by-Step Solution

In this article, we'll walk through the process of evaluating the expression m + 1rac{1}{3} when m = 3rac{1}{2}. This involves substituting the value of $m$ into the expression and simplifying the result. We'll cover the necessary steps to arrive at the solution, expressing it as a fraction, whole number, or mixed number, making it easy to understand for everyone. So, if you're looking to brush up on your algebra skills or need a clear explanation of this problem, you've come to the right place! Let's dive in and get started.

Understanding the Problem: Evaluating Expressions with Mixed Numbers

When it comes to evaluating expressions, it's essential to have a solid grasp of the fundamental concepts. In this particular problem, we are tasked with finding the value of the expression m + 1rac{1}{3} when $m$ is equal to 3rac{1}{2}. This involves substituting the given value of $m$ into the expression and then simplifying the result. The key here is working with mixed numbers, which combine whole numbers and fractions. To effectively solve this, we'll need to convert mixed numbers into improper fractions, perform addition, and then convert back to a mixed number if necessary. This step-by-step approach ensures accuracy and clarity in our solution. Understanding these basics makes the problem much more manageable and sets the stage for tackling similar algebraic challenges. The ability to manipulate fractions and mixed numbers is a core skill in mathematics, applicable in various contexts beyond just algebraic expressions. So, let’s break down each step to ensure a clear understanding of the process.

Step 1: Convert Mixed Numbers to Improper Fractions

To begin, we need to convert the mixed numbers 3rac{1}{2} and 1rac{1}{3} into improper fractions. This is a crucial step because it allows us to perform addition more easily. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Let's start with 3rac{1}{2}. To convert this, we multiply the whole number (3) by the denominator (2) and then add the numerator (1). This gives us $(3 imes 2) + 1 = 7$. We then place this result over the original denominator, giving us rac{7}{2}. Next, we convert 1rac{1}{3} in a similar way. Multiply the whole number (1) by the denominator (3) and add the numerator (1), resulting in $(1 imes 3) + 1 = 4$. Place this over the original denominator to get rac{4}{3}. Now we have both mixed numbers converted into improper fractions, which sets us up perfectly for the next step of adding these fractions together. This conversion is a foundational skill when dealing with mixed numbers in any mathematical operation.

Step 2: Substitute and Rewrite the Expression

Now that we've converted our mixed numbers into improper fractions, the next step is to substitute the value of $m$ into the expression. We know that m = 3rac{1}{2}, which we've converted to rac{7}{2}. Our expression is m + 1rac{1}{3}, and we've converted 1rac{1}{3} to rac{4}{3}. So, substituting the values, our expression becomes rac{7}{2} + rac{4}{3}. This substitution is a fundamental part of evaluating algebraic expressions. By replacing the variable $m$ with its numerical value, we transform the expression into a straightforward addition problem involving fractions. Rewriting the expression in this way makes it much easier to visualize and solve. It also highlights the importance of accurate substitution to ensure the final result is correct. Now that we have our expression in terms of improper fractions, we can move on to the next step, which is finding a common denominator to add these fractions together. This step-by-step approach makes the problem more manageable and reduces the chances of errors.

Step 3: Find a Common Denominator

To add the fractions rac{7}{2} and rac{4}{3}, we need to find a common denominator. A common denominator is a number that both denominators (2 and 3 in this case) can divide into evenly. The easiest way to find a common denominator is to multiply the two denominators together. So, $2 imes 3 = 6$. This means 6 will be our common denominator. Now, we need to convert each fraction to an equivalent fraction with a denominator of 6. For rac{7}{2}, we multiply both the numerator and the denominator by 3 (since $2 imes 3 = 6$), which gives us rac{7 imes 3}{2 imes 3} = rac{21}{6}. For rac{4}{3}, we multiply both the numerator and the denominator by 2 (since $3 imes 2 = 6$), which gives us rac{4 imes 2}{3 imes 2} = rac{8}{6}. Now we have two fractions with the same denominator: rac{21}{6} and rac{8}{6}. Finding a common denominator is a crucial step in adding or subtracting fractions, as it ensures that we are adding or subtracting comparable parts of a whole. With this step complete, we are ready to add the fractions together.

Step 4: Add the Fractions

With our fractions now having a common denominator, we can proceed to add them together. We have rac{21}{6} and rac{8}{6}. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same. So, rac{21}{6} + rac{8}{6} = rac{21 + 8}{6} = rac{29}{6}. This gives us the improper fraction rac{29}{6}. Adding fractions once they have a common denominator is a straightforward process, but it's important to ensure that the numerators are added correctly. Now that we have our result as an improper fraction, the final step is to convert it back into a mixed number, which is often the preferred format for expressing such values. This conversion will give us a clearer understanding of the magnitude of the result and align with the problem's instructions to express the answer as a fraction, whole number, or mixed number.

Step 5: Convert the Improper Fraction to a Mixed Number

Our final step is to convert the improper fraction rac{29}{6} back into a mixed number. To do this, we divide the numerator (29) by the denominator (6). 29 divided by 6 is 4 with a remainder of 5. The quotient (4) becomes the whole number part of our mixed number, the remainder (5) becomes the numerator, and the original denominator (6) remains the same. So, rac{29}{6} converts to 4rac{5}{6}. This mixed number represents the final simplified value of the expression. Converting improper fractions to mixed numbers is essential for expressing results in a clear and understandable format. It allows us to see the whole number part and the fractional part separately, providing a more intuitive sense of the value. This final conversion completes the evaluation of the expression, giving us our answer in the desired format. Now, let's summarize the entire process and highlight the key steps we took to arrive at the solution.

Conclusion: The Final Result

In conclusion, by following a step-by-step approach, we successfully evaluated the expression m + 1rac{1}{3} when m = 3rac{1}{2}. We began by converting the mixed numbers into improper fractions, which allowed us to rewrite the expression as rac{7}{2} + rac{4}{3}. Next, we found a common denominator, converting the fractions to rac{21}{6} and rac{8}{6}. Adding these fractions gave us rac{29}{6}, which we then converted back into the mixed number 4rac{5}{6}. Therefore, the value of the expression m + 1rac{1}{3} when m = 3rac{1}{2} is 4rac{5}{6}. This process demonstrates the importance of understanding how to work with fractions and mixed numbers in algebraic expressions. Each step, from converting mixed numbers to improper fractions to finding a common denominator and converting back to a mixed number, is crucial for arriving at the correct solution. By mastering these skills, you'll be well-equipped to tackle similar problems in algebra and beyond. For further learning and to deepen your understanding of fractions and mixed numbers, you can visit trusted educational resources such as Khan Academy.