Solving Mixed Number Expressions A Step By Step Guide

Navigating the world of mixed numbers can sometimes feel like traversing a mathematical maze, especially when faced with complex expressions involving both addition and subtraction. But fear not, math enthusiasts! This comprehensive guide will break down the expression $\left(4 \frac{5}{6}-5 \frac{3}{4}\right)+\left(2 \frac{1}{2}+6 \frac{1}{3}\right)$ step by step, transforming it from a daunting challenge into a delightful mathematical journey. We'll explore the fundamental principles of mixed number arithmetic, providing you with the tools and techniques to conquer similar problems with confidence and ease. So, let's dive in and unlock the secrets of mixed number calculations!

Understanding Mixed Numbers

Before we tackle the main expression, let's take a moment to solidify our understanding of mixed numbers. Mixed numbers, guys, are a combination of a whole number and a proper fraction (where the numerator is less than the denominator). For example, $4 \frac{5}{6}$ is a mixed number where 4 is the whole number part and $\frac{5}{6}$ is the fractional part. Understanding this fundamental structure is key to performing operations like addition and subtraction. To effectively work with mixed numbers, we often convert them into improper fractions. An improper fraction is where the numerator is greater than or equal to the denominator. Converting mixed numbers to improper fractions makes it much easier to perform arithmetic operations. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction, add the numerator, and then place the result over the original denominator. Let's illustrate this with our first mixed number, $4 \frac{5}{6}$. We multiply the whole number 4 by the denominator 6, which gives us 24. Then, we add the numerator 5, resulting in 29. Finally, we place this over the original denominator 6, giving us the improper fraction $\frac{29}{6}$. Similarly, we can convert $5 \frac{3}{4}$ into an improper fraction. Multiplying 5 by 4 gives us 20, adding 3 gives us 23, and placing this over the denominator 4 results in the improper fraction $\frac{23}{4}$. This conversion process is crucial because it allows us to perform addition and subtraction with fractions that have a common denominator, a concept we'll explore further in the next section. Mastering the conversion between mixed numbers and improper fractions is a foundational skill in arithmetic, and it will significantly simplify more complex calculations involving fractions. By understanding the underlying principles, you can confidently manipulate these numbers and solve a wide range of mathematical problems. Remember, practice makes perfect, so try converting various mixed numbers to improper fractions to reinforce your understanding and build your proficiency.

Tackling the Subtraction: $\left(4 \frac{5}{6}-5 \frac{3}{4}\right)$

Now that we've refreshed our understanding of mixed numbers, let's dive into the first part of our expression: $\left(4 \frac{5}{6}-5 \frac{3}{4}\right)$. As we discussed earlier, the first step is to convert these mixed numbers into improper fractions. We've already converted $4 \frac{5}{6}$ to $\frac{29}{6}$ and $5 \frac{3}{4}$ to $\frac{23}{4}$. So, our expression now looks like this: $\left(\frac{29}{6} - \frac{23}{4}\right)$. To subtract fractions, they need to have a common denominator. This means we need to find the least common multiple (LCM) of 6 and 4. The multiples of 6 are 6, 12, 18, 24, and so on. The multiples of 4 are 4, 8, 12, 16, 20, 24, and so on. The least common multiple of 6 and 4 is 12. So, we need to rewrite both fractions with a denominator of 12. To convert $\frac{29}{6}$ to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 2 (because 6 multiplied by 2 equals 12). This gives us $\frac{29 \times 2}{6 \times 2} = \frac{58}{12}$. Similarly, to convert $\frac{23}{4}$ to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3 (because 4 multiplied by 3 equals 12). This gives us $\frac{23 \times 3}{4 \times 3} = \frac{69}{12}$. Now we can rewrite our subtraction expression as $\left(\frac{58}{12} - \frac{69}{12}\right)$. Subtracting the numerators, we get $58 - 69 = -11$. So, the result of the subtraction is $\frac{-11}{12}$. This result is a negative fraction, which is perfectly valid. It simply indicates that the second fraction we subtracted was larger than the first. Understanding how to find the least common multiple and rewrite fractions with a common denominator is a crucial skill in fraction arithmetic. It allows us to perform addition and subtraction accurately and efficiently. By breaking down the problem into smaller steps, we can systematically solve even complex expressions involving fractions.

Adding the Fractions: $\left(2 \frac{1}{2}+6 \frac{1}{3}\right)$

Let's move on to the second part of our expression: $\left(2 \frac{1}{2}+6 \frac{1}{3}\right)$. Just like with subtraction, our first step in adding mixed numbers is to convert them into improper fractions. Converting $2 \frac{1}{2}$, we multiply 2 by 2 (the denominator) which gives us 4, then add 1 (the numerator) to get 5. Placing this over the denominator 2, we get the improper fraction $\frac{5}{2}$. For $6 \frac{1}{3}$, we multiply 6 by 3, which gives us 18, add 1 to get 19, and place this over the denominator 3, resulting in the improper fraction $\frac{19}{3}$. Now our expression looks like this: $\left(\frac{5}{2} + \frac{19}{3}\right)$. To add these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. So, we need to rewrite both fractions with a denominator of 6. To convert $\frac{5}{2}$ to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 3 (because 2 multiplied by 3 equals 6). This gives us $\frac{5 \times 3}{2 \times 3} = \frac{15}{6}$. To convert $\frac{19}{3}$ to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 2 (because 3 multiplied by 2 equals 6). This gives us $\frac{19 \times 2}{3 \times 2} = \frac{38}{6}$. Now we can rewrite our addition expression as $\left(\frac{15}{6} + \frac{38}{6}\right)$. Adding the numerators, we get $15 + 38 = 53$. So, the result of the addition is $\frac{53}{6}$. This is an improper fraction, and we can convert it back to a mixed number if desired. To do this, we divide 53 by 6. 6 goes into 53 eight times (8 x 6 = 48) with a remainder of 5. So, $\frac{53}{6}$ is equal to $8 \frac{5}{6}$. Mastering the addition of fractions, just like subtraction, involves understanding the concept of common denominators and the ability to convert between improper fractions and mixed numbers. These skills are essential for working with fractions confidently and accurately.

Putting It All Together: $\left(\frac{-11}{12}\right) + \left(\frac{53}{6}\right)$

Now, let's bring the two parts of our problem together. We found that $\left(4 \frac{5}{6}-5 \frac{3}{4}\right) = \frac{-11}{12}$ and $\left(2 \frac{1}{2}+6 \frac{1}{3}\right) = \frac{53}{6}$. So, our original expression is now simplified to $\frac{-11}{12} + \frac{53}{6}$. To add these fractions, we once again need a common denominator. The least common multiple of 12 and 6 is 12. The first fraction, $\frac{-11}{12}$, already has a denominator of 12, so we don't need to change it. To convert $\frac{53}{6}$ to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 2 (because 6 multiplied by 2 equals 12). This gives us $\frac{53 \times 2}{6 \times 2} = \frac{106}{12}$. Now we can rewrite our expression as $\frac{-11}{12} + \frac{106}{12}$. Adding the numerators, we get $-11 + 106 = 95$. So, the result of the addition is $\frac{95}{12}$. This is an improper fraction, and we can convert it back to a mixed number. Dividing 95 by 12, we find that 12 goes into 95 seven times (7 x 12 = 84) with a remainder of 11. Therefore, $\frac{95}{12}$ is equal to $7 \frac{11}{12}$. So, the final answer to our expression $\left(4 \frac{5}{6}-5 \frac{3}{4}\right)+\left(2 \frac{1}{2}+6 \frac{1}{3}\right)$ is $7 \frac{11}{12}$. By breaking down the problem into manageable steps – converting mixed numbers to improper fractions, finding common denominators, performing addition and subtraction, and converting back to mixed numbers if necessary – we successfully navigated this complex expression. This systematic approach is the key to mastering mixed number arithmetic.

Key Takeaways and Practice

Congratulations, guys! You've made it through a detailed exploration of mixed number arithmetic. Let's recap the key takeaways from this journey:

• Convert Mixed Numbers to Improper Fractions: This is the crucial first step for simplifying addition and subtraction. Remember the formula: (Whole Number x Denominator) + Numerator / Denominator.
• Find the Least Common Multiple (LCM): To add or subtract fractions, they must have a common denominator. The LCM is the smallest number that is a multiple of both denominators.
• Rewrite Fractions with the Common Denominator: Multiply the numerator and denominator of each fraction by the appropriate factor to achieve the common denominator.
• Perform Addition or Subtraction: Once the fractions have a common denominator, simply add or subtract the numerators.
• Simplify and Convert Back to Mixed Numbers (if needed): If the result is an improper fraction, convert it back to a mixed number for easier interpretation.

To solidify your understanding, practice is essential. Try solving similar expressions with different mixed numbers. You can even create your own problems and challenge yourself! The more you practice, the more confident and proficient you'll become in working with mixed numbers. Remember, mathematics is like learning a new language; it takes time and dedication to become fluent. So, keep practicing, keep exploring, and keep challenging yourself. With each problem you solve, you'll build a stronger foundation and unlock new levels of mathematical understanding. Embrace the journey, and enjoy the process of mastering mixed number arithmetic!