Addition and Subtraction of Mixed Numbers: The common denominator is smaller than the product of the denominators

To solve the problem of adding $3\frac{1}{2} + 8\frac{3}{4}$, we'll go through these steps:

• Step 1: Separate and add the whole numbers.
• Step 2: Add the fractional parts by finding a common denominator.
• Step 3: Combine the results and simplify if necessary.

Let's apply these steps in detail:

Step 1: Add the whole numbers:
$3 + 8 = 11$.

Step 2: Add the fractions $\frac{1}{2}$ and $\frac{3}{4}$:
The denominators are 2 and 4. The least common denominator is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 4:
$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Now, add the fractions:
$\frac{2}{4} + \frac{3}{4} = \frac{2+3}{4} = \frac{5}{4}$.

$\frac{5}{4}$ is an improper fraction, which can be converted to a mixed number:
$\frac{5}{4} = 1\frac{1}{4}$.

Step 3: Combine the whole numbers and the mixed fraction:
$11 + 1\frac{1}{4} = 12\frac{1}{4}$.

Therefore, the solution to the problem is $12\frac{1}{4}$.

To find the sum of $2\frac{1}{6} + 2\frac{2}{3}$, we will follow these steps:

• Step 1: Separate the mixed numbers into their whole and fractional parts:
• Whole parts: $2$ and $2$.
• Fractional parts: $\frac{1}{6}$ and $\frac{2}{3}$.
• Step 2: Add the whole parts:
• $2 + 2 = 4$.
• Step 3: Convert fractional parts using a common denominator:
• The fractions are $\frac{1}{6}$ and $\frac{2}{3}$.
• The least common denominator of $6$ and $3$ is $6$.
• $\frac{2}{3}$ is converted to $\frac{4}{6}$ by multiplying both numerator and denominator by $2$.
• Step 4: Add the fractions:
• $\frac{1}{6} + \frac{4}{6} = \frac{5}{6}$.
• Step 5: Combine results:
• Combine the whole and fractional parts: $4 + \frac{5}{6} = 4\frac{5}{6}$.

Therefore, the sum of the mixed numbers $2\frac{1}{6}$ and $2\frac{2}{3}$ is $4\frac{5}{6}$.

To solve the problem of adding $7\frac{2}{4} + 3\frac{1}{4}$, we follow these steps:

• Step 1: Add the whole numbers from both mixed numbers: $7 + 3 = 10$.
• Step 2: Add the fractional parts. Since the denominators are the same, we simply add the numerators: $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.
• Step 3: Combine the results: The sum of the whole numbers is $10$ and the sum of the fractions is $\frac{3}{4}$. Therefore, the total is $10\frac{3}{4}$.

The final result of the mixed number addition $7\frac{2}{4} + 3\frac{1}{4}$ is $10\frac{3}{4}$.

To solve this problem, we'll follow these steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Find a common denominator for the fractions.
• Step 3: Add the fractions and simplify the result.
• Step 4: Convert the result back to a mixed number if necessary.

Now, let's work through each step:

Step 1: Convert mixed numbers to improper fractions.

The first mixed number is $6\frac{3}{8}$. To convert this:

• The whole number part is 6, and the fractional part is $\frac{3}{8}$.
• Multiply the whole number by the denominator: $6 \times 8 = 48$.
• Add the numerator: $48 + 3 = 51$.
• The improper fraction is $\frac{51}{8}$.

The second mixed number is $2\frac{3}{4}$. To convert this:

• The whole number part is 2, and the fractional part is $\frac{3}{4}$.
• Multiply the whole number by the denominator: $2 \times 4 = 8$.
• Add the numerator: $8 + 3 = 11$.
• The improper fraction is $\frac{11}{4}$.

Step 2: Find a common denominator for $\frac{51}{8}$ and $\frac{11}{4}$.

The denominators are 8 and 4. The least common denominator is 8.

Step 3: Add the fractions.

Convert $\frac{11}{4}$ to the equivalent fraction with a denominator of 8:

• Multiply both the numerator and the denominator by 2: $\frac{11 \times 2}{4 \times 2} = \frac{22}{8}$.

Now add: $\frac{51}{8} + \frac{22}{8} = \frac{51 + 22}{8} = \frac{73}{8}$.

Step 4: Convert $\frac{73}{8}$ back to a mixed number.

• Divide 73 by 8: $73 \div 8 = 9$ remainder 1.
• The quotient is 9, and the remainder is 1.
• The mixed number is $9\frac{1}{8}$.

Therefore, the solution to the problem is $9\frac{1}{8}$.

Let's solve the problem by adding the mixed numbers step-by-step:

• Step 1: Add the whole numbers. The whole numbers are $3$ and $1$, so $3 + 1 = 4$.
• Step 2: Add the fractional parts. The fractions are $\frac{1}{2}$ and $\frac{2}{6}$.
• Step 3: Find a common denominator for the fractions. The denominators are $2$ and $6$. The least common denominator (LCD) is $6$.
• Step 4: Convert $\frac{1}{2}$ to a fraction with the common denominator: $\frac{1}{2} = \frac{3}{6}$.
• Step 5: The fractions $\frac{3}{6}$ and $\frac{2}{6}$ can now be added: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.
• Step 6: Combine the results from Steps 1 and 5: $4 + \frac{5}{6} = 4\frac{5}{6}$.

Therefore, the result of adding $3\frac{1}{2} + 1\frac{2}{6}$ is $4\frac{5}{6}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the integer and fractional parts of each mixed number.
• Step 2: Find a common denominator for the fractional parts.
• Step 3: Add the fractional parts.
• Step 4: Add the integer parts.
• Step 5: Combine the results from Steps 3 and 4.

Now, let's work through each step:

Step 1: Identify the integer and fractional parts:
$13\frac{3}{10}$ has an integer part of 13 and a fractional part of $\frac{3}{10}$.
$2\frac{1}{2}$ has an integer part of 2 and a fractional part of $\frac{1}{2}$.

Step 2: Find a common denominator for $\frac{3}{10}$ and $\frac{1}{2}$.
The denominators are 10 and 2, with the common denominator being 10. Convert $\frac{1}{2}$ to an equivalent fraction with this common denominator: $\frac{1}{2} = \frac{5}{10}$.

Step 3: Add the fractional parts:
$\frac{3}{10} + \frac{5}{10} = \frac{8}{10}$.

Step 4: Add the integer parts:
$13 + 2 = 15$.

Step 5: Combine the integer sum and fractional sum:
The result is $15\frac{8}{10}$.

Therefore, the solution to the problem is $15\frac{8}{10}$.

Let's solve the problem by converting the mixed numbers into improper fractions, finding a common denominator, and adding the fractions.

Step 1: Convert Mixed Numbers to Improper Fractions
- $2\frac{7}{9}$ becomes $\frac{2 \times 9 + 7}{9} = \frac{18 + 7}{9} = \frac{25}{9}$.
- $1\frac{1}{3}$ becomes $\frac{1 \times 3 + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3}$.

Step 2: Find a Common Denominator
- The denominators are 9 and 3. The least common denominator is 9.

Step 3: Convert Fractions to a Common Denominator
- $\frac{25}{9}$ is already with the denominator 9.
- $\frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9}$.

Step 4: Add the Fractions
$\frac{25}{9} + \frac{12}{9} = \frac{25 + 12}{9} = \frac{37}{9}$.

Step 5: Convert the Improper Fraction Back to a Mixed Number
- $\frac{37}{9}$ is $4 \frac{1}{9}$ because 37 divided by 9 is 4 with a remainder of 1.

Therefore, the solution to the problem $2\frac{7}{9}+1\frac{1}{3}$ is $4\frac{1}{9}$.

To solve the problem $6\frac{5}{14} - 2\frac{1}{7}$, we'll follow these steps:

• Step 1: Identify a common denominator for the fractions. The denominators are 14 and 7, and the least common denominator is 14.
• Step 2: Convert $2\frac{1}{7}$ to a fraction with a denominator of 14 to match $6\frac{5}{14}$.
• Step 3: Subtract the fractions.
• Step 4: Subtract the whole numbers.

Now, let's work through each step:

Step 1: The fractions are $\frac{5}{14}$ and $\frac{1}{7}$. The common denominator is 14.

Step 2: Convert $2\frac{1}{7}$: Start by converting $\frac{1}{7}$ to a denominator of 14:
$\frac{1}{7}$ is equivalent to $\frac{2}{14}$ (since $1 \times 2 = 2$ and $7 \times 2 = 14$).
So, $2\frac{1}{7} = 2\frac{2}{14}$.

Step 3: Subtract the fractions:
Subtract $\frac{2}{14}$ from $\frac{5}{14}$:
$\frac{5}{14} - \frac{2}{14} = \frac{3}{14}$.

Step 4: Subtract the whole numbers:
$6 - 2 = 4$.

Therefore, the final result is:
$4\frac{3}{14}$.

The correct answer to the problem $6\frac{5}{14} - 2\frac{1}{7}$ is $4\frac{3}{14}$.

To solve this problem, we will subtract the mixed numbers $3\frac{7}{15}$ and $2\frac{1}{5}$ by considering their whole number parts and fractions separately.

Step 1: Subtract the Whole Numbers
The whole number part of the first mixed number is 3, and for the second mixed number, it is 2. Subtracting these gives:

Step 2: Subtract the Fractional Parts
The fractional part of the first number is $\frac{7}{15}$, and for the second, it is $\frac{1}{5}$. We need a common denominator to subtract these fractions. The least common denominator for 15 and 5 is 15. Convert $\frac{1}{5}$ to a fraction with denominator 15:

Now, subtract the fractions:

Step 3: Combine the Whole Number and Fractional Parts
The whole number part is 1, and the fractional part is $\frac{4}{15}$. Combining these gives:

Thus, the result of the subtraction $3\frac{7}{15} - 2\frac{1}{5}$ is:

$1\frac{4}{15}$

Therefore, the solution to the problem is $1\frac{4}{15}$.

Let's work through the problem:

Step 1: Convert to improper fractions.
The mixed number $6\frac{1}{2}$ is converted to an improper fraction as follows:

• Multiply the whole number 6 by the denominator 2: $6 \times 2 = 12$.

• Add the numerator 1: $12 + 1 = 13$.

• The improper fraction is $\frac{13}{2}$.

The mixed number $1\frac{3}{4}$ is converted to an improper fraction as follows:

• Multiply the whole number 1 by the denominator 4: $1 \times 4 = 4$.

• Add the numerator 3: $4 + 3 = 7$.

• The improper fraction is $\frac{7}{4}$.

Step 2: Find a common denominator.
The denominators are 2 and 4. The least common denominator is 4.

Step 3: Adjust fractions to have the common denominator.
$\frac{13}{2}$ needs to be adjusted to have a denominator of 4:

• Multiply both the numerator and the denominator by 2: $\frac{13 \times 2}{2 \times 2} = \frac{26}{4}$.

Step 4: Perform the subtraction.
Subtract $\frac{7}{4}$ from $\frac{26}{4}$:

• $\frac{26}{4} - \frac{7}{4} = \frac{26 - 7}{4} = \frac{19}{4}$.

Step 5: Convert the improper fraction back to a mixed number.
Divide the numerator by the denominator:

• $19 \div 4 = 4$ with a remainder of 3.

• The mixed number is $4\frac{3}{4}$.

Therefore, the solution to the problem is $4\frac{3}{4}$.