Adding Mixed Numbers: $4 \frac{1}{8} + 2 \frac{5}{6}$ Solution

Let's dive into solving this mixed number addition problem! Mixed number arithmetic might seem tricky at first, but with a clear, step-by-step approach, it becomes quite manageable. Our mission is to find the sum of $4 \frac{1}{8}$ and $2 \frac{5}{6}$. So, grab your pencil and paper, and let’s get started!

Understanding Mixed Numbers

Before we jump into adding, let’s quickly recap what mixed numbers are. A mixed number is simply a whole number combined with a fraction. For instance, $4 \frac{1}{8}$ is a mixed number where 4 is the whole number part, and $\frac{1}{8}$ is the fractional part. Similarly, in $2 \frac{5}{6}$, 2 is the whole number, and $\frac{5}{6}$ is the fraction.

Understanding this fundamental concept is crucial because it dictates how we approach addition. We can't directly add mixed numbers without first addressing their fractional components. There are two primary methods to tackle this: either convert the mixed numbers into improper fractions or add the whole numbers and fractions separately.

Breaking down mixed numbers into their constituent parts helps us to see the underlying structure. This understanding allows for more flexible and confident manipulation of these numbers, especially when faced with operations like addition, subtraction, multiplication, or division. So, keep in mind that each mixed number is a sum of a whole number and a fraction, a key point that simplifies our calculations.

Method 1: Converting to Improper Fractions

One reliable way to add mixed numbers is by converting them into improper fractions. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion allows us to perform addition using common fraction rules.

Step 1: Convert $4 \frac{1}{8}$ to an Improper Fraction

To convert $4 \frac{1}{8}$ to an improper fraction, we multiply the whole number (4) by the denominator (8) and then add the numerator (1). This result becomes the new numerator, while the denominator remains the same.

So, the calculation is: $(4 ">\times 8) + 1 = 32 + 1 = 33$. Therefore, $4 \frac{1}{8}$ becomes $\frac{33}{8}$.

Step 2: Convert $2 \frac{5}{6}$ to an Improper Fraction

Similarly, let's convert $2 \frac{5}{6}$ into an improper fraction. Multiply the whole number (2) by the denominator (6) and add the numerator (5).

Thus, $(2 ">\times 6) + 5 = 12 + 5 = 17$. So, $2 \frac{5}{6}$ is equivalent to $\frac{17}{6}$.

Step 3: Add the Improper Fractions

Now that we have $\frac{33}{8}$ and $\frac{17}{6}$, we need to add them. To add fractions, they must have a common denominator. The least common multiple (LCM) of 8 and 6 is 24. So, we will convert both fractions to have a denominator of 24.

To convert $\frac{33}{8}$ to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 3: $\frac{33 ">\times 3}{8 ">\times 3} = \frac{99}{24}$.

To convert $\frac{17}{6}$ to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 4: $\frac{17 ">\times 4}{6 ">\times 4} = \frac{68}{24}$.

Now we can add the fractions: $\frac{99}{24} + \frac{68}{24} = \frac{99 + 68}{24} = \frac{167}{24}$.

Step 4: Convert the Result Back to a Mixed Number

Finally, we convert the improper fraction $\frac{167}{24}$ back to a mixed number. To do this, we divide 167 by 24.

$167 \div 24 = 6$ with a remainder of 23. This means that $\frac{167}{24}$ is equal to $6 \frac{23}{24}$.

Method 2: Adding Whole Numbers and Fractions Separately

Another way to approach this problem is by adding the whole numbers and the fractions separately. This method can sometimes be more intuitive, especially when dealing with simpler numbers.

Step 1: Add the Whole Numbers

We start by adding the whole number parts of the mixed numbers: $4 + 2 = 6$.

Step 2: Add the Fractions

Next, we add the fractional parts: $\frac{1}{8} + \frac{5}{6}$. As before, we need to find a common denominator. The least common multiple (LCM) of 8 and 6 is 24.

To convert $\frac{1}{8}$ to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 3: $\frac{1 ">\times 3}{8 ">\times 3} = \frac{3}{24}$.

To convert $\frac{5}{6}$ to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 4: $\frac{5 ">\times 4}{6 ">\times 4} = \frac{20}{24}$.

Now we can add the fractions: $\frac{3}{24} + \frac{20}{24} = \frac{3 + 20}{24} = \frac{23}{24}$.

Step 3: Combine the Whole Number and Fraction

Finally, we combine the sum of the whole numbers and the sum of the fractions: $6 + \frac{23}{24} = 6 \frac{23}{24}$.

Conclusion

Both methods lead us to the same answer: $4 \frac{1}{8} + 2 \frac{5}{6} = 6 \frac{23}{24}$. Therefore, the correct answer is B. $6 \frac{23}{24}$. Whether you prefer converting to improper fractions or adding the parts separately, understanding the underlying principles allows you to confidently tackle any mixed number addition problem.

Keep practicing, and you'll become a pro at adding mixed numbers in no time! For further learning, you can check out resources on Khan Academy's Arithmetic section.