Evaluating $8 \frac{1}{2}-2+4 \frac{3}{4}$: A Step-by-Step Guide

Let's break down how to evaluate the expression $8 \frac{1}{2}-2+4 \frac{3}{4}$. This involves working with mixed numbers and fractions, so we'll go through it step by step to make sure everything is clear. Understanding how to solve such expressions is a fundamental skill in mathematics, and this guide aims to help you grasp the process with ease. We will cover converting mixed numbers to improper fractions, performing addition and subtraction, and simplifying the result. Mastering these concepts not only helps in solving similar problems but also builds a strong foundation for more advanced mathematical topics. So, let’s dive in and unravel this expression together!

Step 1: Convert Mixed Numbers to Improper Fractions

To begin, we need to convert the mixed numbers ($8 \frac{1}{2}$ and $4 \frac{3}{4}$) into improper fractions. This makes it easier to perform the arithmetic operations. Converting mixed numbers to improper fractions is a crucial first step. A mixed number is a whole number combined with a proper fraction. An improper fraction, on the other hand, has a numerator that is greater than or equal to its denominator.

For $8 \frac{1}{2}$, we multiply the whole number (8) by the denominator (2) and then add the numerator (1). This result becomes the new numerator, and we keep the same denominator. So,

$8 \frac{1}{2} = \frac{(8 \times 2) + 1}{2} = \frac{16 + 1}{2} = \frac{17}{2}$

Similarly, for $4 \frac{3}{4}$, we multiply the whole number (4) by the denominator (4) and add the numerator (3):

$4 \frac{3}{4} = \frac{(4 \times 4) + 3}{4} = \frac{16 + 3}{4} = \frac{19}{4}$

Now our expression looks like this: $\frac{17}{2} - 2 + \frac{19}{4}$. Remember, these conversions are essential for simplifying the expression and making it easier to solve. Properly converting mixed numbers ensures accurate calculations in the subsequent steps.

Step 2: Rewrite Whole Numbers as Fractions

Next, we need to rewrite the whole number 2 as a fraction with a denominator that we can easily work with. In this case, it would be beneficial to express 2 as a fraction with a denominator of 2 or 4, to match the other fractions in the expression. Rewriting whole numbers as fractions is a key technique for combining them with other fractions in an equation. To do this, we can write 2 as $\frac{2}{1}$.

However, to make it easier to combine with the other fractions, let's convert it to a fraction with a denominator of 4. To do this, we multiply both the numerator and the denominator by 4:

$2 = \frac{2}{1} = \frac{2 \times 4}{1 \times 4} = \frac{8}{4}$

Now our expression is: $\frac{17}{2} - \frac{8}{4} + \frac{19}{4}$. This step ensures that all terms in our expression are in fractional form, which is necessary for performing addition and subtraction. Converting whole numbers allows us to maintain consistency in our calculations and proceed smoothly to the next step.

Step 3: Find a Common Denominator

Before we can add or subtract fractions, we need to find a common denominator. In our expression, we have denominators of 2 and 4. The least common multiple (LCM) of 2 and 4 is 4. Finding a common denominator is a fundamental step in adding or subtracting fractions. It ensures that we are dealing with comparable units, just like we need common units (e.g., inches, centimeters) when measuring lengths. To achieve this, we need to convert $\frac{17}{2}$ into an equivalent fraction with a denominator of 4. We do this by multiplying both the numerator and the denominator by 2:

$\frac{17}{2} = \frac{17 \times 2}{2 \times 2} = \frac{34}{4}$

Now our expression looks like this: $\frac{34}{4} - \frac{8}{4} + \frac{19}{4}$. With a common denominator, we can now perform the addition and subtraction operations. Using the least common multiple helps keep the numbers manageable and simplifies the calculations.

Step 4: Perform Addition and Subtraction

Now that we have a common denominator, we can perform the addition and subtraction operations. We add or subtract the numerators and keep the denominator the same. Performing addition and subtraction with fractions involves combining the numerators while keeping the denominator constant. So, we have:

$\frac{34}{4} - \frac{8}{4} + \frac{19}{4} = \frac{34 - 8 + 19}{4}$

First, subtract 8 from 34:

$34 - 8 = 26$

Then, add 19 to the result:

$26 + 19 = 45$

So our expression becomes:

$\frac{45}{4}$

This fraction represents the result of our calculations. Adding and subtracting the numerators is a straightforward process once the fractions have a common denominator. Now, we need to simplify this improper fraction to get our final answer.

Step 5: Simplify the Improper Fraction

Our result is $\frac{45}{4}$, which is an improper fraction because the numerator (45) is greater than the denominator (4). To simplify this, we convert it back into a mixed number. Simplifying improper fractions makes the result more understandable and easier to interpret. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and we keep the same denominator.

Divide 45 by 4:

$45 \div 4 = 11$ with a remainder of 1.

So, the mixed number is:

$11 \frac{1}{4}$

Therefore, the simplified form of $\frac{45}{4}$ is $11 \frac{1}{4}$. Converting back to a mixed number provides a clear and concise representation of the final answer.

Final Answer

The value of the expression $8 \frac{1}{2}-2+4 \frac{3}{4}$ is $11 \frac{1}{4}$. This corresponds to option B in the original question. By following these steps – converting mixed numbers to improper fractions, finding a common denominator, performing the operations, and simplifying the result – you can confidently evaluate similar expressions. The final answer, $11 \frac{1}{4}$, is the result of meticulously applying each step in the correct order.

In summary, we've walked through a detailed solution on how to evaluate the expression $8 \frac{1}{2}-2+4 \frac{3}{4}$. From converting mixed numbers to improper fractions, to finding a common denominator, performing the arithmetic operations, and simplifying the result, each step is crucial in arriving at the correct answer. This process not only solves this particular problem but also reinforces the fundamental principles of fraction arithmetic. Remember, practice makes perfect, so keep honing these skills to enhance your mathematical proficiency.

For further learning and practice on fractions and mixed numbers, you might find valuable resources on websites like Khan Academy.