Addition and Subtraction of Mixed Numbers: One of the denominators is the common denominator

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

• Step 1: Convert $1\frac{3}{4}$ to a fraction with a denominator of 8.
• Step 2: Subtract the whole numbers and fractions separately.
• Step 3: Simplify the resulting fraction and combine it with the whole number part.

Now, let's work through each step:

Step 1: Convert $1\frac{3}{4}$ to a fraction with a denominator of 8.
Since $\frac{3}{4} = \frac{6}{8}$ (by multiplying numerator and denominator by 2), we have:
$1\frac{3}{4} = 1 + \frac{6}{8} = \frac{8}{8} + \frac{6}{8} = \frac{14}{8}$.

Step 2: Subtract the whole numbers and fractions separately.
Original problem: $3\frac{2}{8} - 1\frac{3}{4} = \frac{26}{8} - \frac{14}{8}$.

Subtract the fractions:
$\frac{26}{8} - \frac{14}{8} = \frac{12}{8}$.

Step 3: Simplify the resulting fraction and combine it with the whole number part.
$\frac{12}{8} = \frac{3}{2}$.

Therefore, after subtracting and simplifying, we find that:

Therefore, the solution to the problem is $1\frac{1}{2}$.

To solve the problem $2\frac{4}{3} - 1\frac{1}{2}$, let's follow these steps:

• Step 1: Convert the mixed numbers into improper fractions.
• Step 2: Find a common denominator for the fractions.
• Step 3: Perform the subtraction of the fractions.
• Step 4: Simplify the resulting fraction if possible, and convert it back to a mixed number if needed.

Now let's work through these steps:

Step 1: Convert the mixed numbers into improper fractions.

$2\frac{4}{3} = \frac{2 \times 3 + 4}{3} = \frac{6 + 4}{3} = \frac{10}{3}$

$1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}$

Step 2: Identify a common denominator. The denominators are 3 and 2; the least common denominator is 6.

Convert $\frac{10}{3}$ to have a denominator of 6:

$\frac{10}{3} \times \frac{2}{2} = \frac{20}{6}$

Convert $\frac{3}{2}$ to have a denominator of 6:

$\frac{3}{2} \times \frac{3}{3} = \frac{9}{6}$

Step 3: Perform the subtraction:

$\frac{20}{6} - \frac{9}{6} = \frac{20 - 9}{6} = \frac{11}{6}$

Step 4: Convert $\frac{11}{6}$ back to a mixed number:

$\frac{11}{6} = 1\frac{5}{6}$

Therefore, the solution to the problem is $1\frac{5}{6}$.

To solve the addition of the mixed numbers $4\frac{5}{6}$ and $1\frac{1}{12}$, follow these steps:

• Step 1: Add the whole numbers.
  The whole numbers are 4 and 1, so we add these together:
  $4 + 1 = 5$.
• Step 2: Add the fractional parts.
  First, find a common denominator for the fractions $\frac{5}{6}$ and $\frac{1}{12}$. The least common denominator of 6 and 12 is 12.
  Convert $\frac{5}{6}$ to twelfths: $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$.
  Now add the fractions: $\frac{10}{12} + \frac{1}{12} = \frac{11}{12}$.
• Step 3: Combine the results.
  Add the whole number and the sum of the fractions:
  $5 + \frac{11}{12} = 5\frac{11}{12}$.

Thus, the sum of $4\frac{5}{6} + 1\frac{1}{12}$ is $5\frac{11}{12}$.

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

• Step 1: Convert the mixed numbers to improper fractions.
• Step 2: Find a common denominator and perform the addition.
• Step 3: Convert the resulting improper fraction back to a mixed number.

Now, let's work through each step:
Step 1: Convert each mixed number to an improper fraction:

• $1\frac{1}{4}$ becomes $\frac{5}{4}$ because $1 \times 4 + 1 = 5$.
• $2\frac{1}{2}$ becomes $\frac{5}{2}$ because $2 \times 2 + 1 = 5$.

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

To solve this problem, we will proceed with the following steps:

• Step 1: Identify the whole and fractional parts of each mixed number.

• Step 2: Convert the fractions to have a common denominator.

• Step 3: Add the fractional parts.

• Step 4: Add the whole numbers.

• Step 5: Combine the results to obtain the total.

Now, let's go through the solution:

Step 1: Separate the mixed numbers into whole and fractional parts:

• $1\frac{2}{3} = 1 + \frac{2}{3}$

• $1\frac{2}{6} = 1 + \frac{2}{6}$

Step 2: Convert the fractions $\frac{2}{3}$ and $\frac{2}{6}$ to have a common denominator. The least common multiple of 3 and 6 is 6.

Convert $\frac{2}{3}$:

$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$

Step 3: Add the fractional parts:

$\frac{4}{6} + \frac{2}{6} = \frac{4+2}{6} = \frac{6}{6} = 1$

Step 4: Add the whole number parts:

$1 + 1 = 2$

Step 5: Combine the whole and fractional results:

Adding the whole sum and fractional sum gives: $2 + 1 = 3$

Therefore, the sum of $1\frac{2}{3} + 1\frac{2}{6}$ is $3$.

To solve this problem, we will follow a structured approach to subtract the given mixed numbers:

• Step 1: Convert mixed numbers to improper fractions.

• Step 2: Find a common denominator for the fractions.

• Step 3: Subtract the improper fractions.

• Step 4: Convert the result back to a mixed number if necessary.

Let's begin with each step in detail:
Step 1: Convert each mixed number to an improper fraction.
- $6\frac{1}{9}$ becomes $\frac{55}{9}$ because $6 \times 9 + 1 = 55$.
- $2\frac{2}{3}$ becomes $\frac{8}{3}$ because $2 \times 3 + 2 = 8$.

Step 2: Convert $\frac{8}{3}$ to have a common denominator with $\frac{55}{9}$.
The least common denominator (LCD) is 9.
Multiply the numerator and the denominator of $\frac{8}{3}$ by 3 to get $\frac{24}{9}$.

Step 3: Subtract the improper fractions:
$\frac{55}{9} - \frac{24}{9} = \frac{31}{9}$.

Step 4: Convert the improper fraction back to a mixed number.
$\frac{31}{9}$ simplifies to $3\frac{4}{9}$, because $31 \div 9$ equals 3 with a remainder of 4.

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

To solve the problem $3\frac{3}{5} + 1\frac{1}{15}$, we'll follow these steps:

• Step 1: Convert each mixed number to an improper fraction.
  - For $3\frac{3}{5}$: Multiply the whole number $3$ by the denominator $5$ and add the numerator $3$: $(3 \times 5) + 3 = 15 + 3 = 18$, so we have $\frac{18}{5}$.
  - For $1\frac{1}{15}$: Multiply the whole number $1$ by the denominator $15$ and add the numerator $1$: $(1 \times 15) + 1 = 15 + 1 = 16$, so we have $\frac{16}{15}$.
• Step 2: Find the least common denominator (LCD).
  - The denominators are $5$ and $15$, the LCD is $15$.
• Step 3: Convert fractions to have the same denominator.
  - $\frac{18}{5} = \frac{18 \times 3}{5 \times 3} = \frac{54}{15}$
  - $\frac{16}{15}$ is already expressed with the denominator $15$.
• Step 4: Add the fractions.
  - $\frac{54}{15} + \frac{16}{15} = \frac{54 + 16}{15} = \frac{70}{15}$
• Step 5: Simplify the fraction.
  - Simplify $\frac{70}{15}$. The greatest common divisor of $70$ and $15$ is $5$. So, $\frac{70}{15} = \frac{70 \div 5}{15 \div 5} = \frac{14}{3}$.
• Step 6: Convert the improper fraction back to a mixed number.
  - $\frac{14}{3} = 4\frac{2}{3}$ (since $14 \div 3 = 4$ remainder $2$)

Therefore, the sum of $3\frac{3}{5} + 1\frac{1}{15}$ is $\mathbf{4\frac{2}{3}}$.

To solve the problem of adding $12\frac{1}{49} + 2\frac{3}{7}$, we'll follow step-by-step procedures:

• Step 1: Identify the whole numbers and fractions:
  Whole numbers: 12 and 2.
  Fractions: $\frac{1}{49}$ and $\frac{3}{7}$.
• Step 2: Convert $\frac{3}{7}$ to a fraction with a denominator of 49:
  Multiply both the numerator and denominator by 7: $\frac{3 \times 7}{7 \times 7} = \frac{21}{49}$.
• Step 3: Add the fractions:
  $\frac{1}{49} + \frac{21}{49} = \frac{1 + 21}{49} = \frac{22}{49}$.
• Step 4: Add the whole numbers:
  12 + 2 = 14.
• Step 5: Combine the whole number result and the fraction result:
  The result is $14\frac{22}{49}$.

Therefore, the solution to the problem $12\frac{1}{49} + 2\frac{3}{7}$ is $14\frac{22}{49}$, which matches choice 4.

To solve this problem, follow these steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Find a common denominator for the fractions.
• Step 3: Subtract the fractions.
• Step 4: Simplify the result if needed.

Let's begin with Step 1:
Convert $3\frac{1}{3}$ to an improper fraction:
$3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}$
Convert $2\frac{14}{15}$ to an improper fraction:
$2\frac{14}{15} = \frac{2 \times 15 + 14}{15} = \frac{44}{15}$

Step 2: Find a common denominator for $\frac{10}{3}$ and $\frac{44}{15}$.
The least common multiple of 3 and 15 is 15, so we'll use 15 as the common denominator.

Step 3: Convert $\frac{10}{3}$ to have the denominator of 15:
$\frac{10}{3} = \frac{10 \times 5}{3 \times 5} = \frac{50}{15}$

Now, perform the subtraction:
$\frac{50}{15} - \frac{44}{15} = \frac{50 - 44}{15} = \frac{6}{15}$

Step 4: Simplify $\frac{6}{15}$.
The greatest common divisor of 6 and 15 is 3, so divide the numerator and denominator by 3:
$\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}$

Thus, the final answer is $\frac{2}{5}$.

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

• Step 1: Convert fractions to a common denominator.
• Step 2: Add the fractions.
• Step 3: Add the whole numbers and the resulting fraction.
• Step 4: Simplify the result.

Now, let's work through each step:

Step 1: Convert $\frac{5}{8}$ to a fraction with denominator 16.
Currently, $\frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16}$.

Step 2: Add the fractions $\frac{5}{16}$ and $\frac{10}{16}$.
$\frac{5}{16} + \frac{10}{16} = \frac{15}{16}$.

Step 3: Add the whole numbers of the mixed numbers.
The whole numbers are 3 and 6, giving $3 + 6 = 9$.

Step 4: Combine the results of steps 2 and 3 into a mixed number.
$9 + \frac{15}{16} = 9\frac{15}{16}$.

Therefore, the solution to the problem is $9\frac{15}{16}$, which corresponds to choice 3 in the provided options.

Note that the right-hand side of the addition exercise between the fractions gives a result of a whole number, so we'll start with that:

$6\frac{2}{3}+\frac{1}{3}=7$

Giving us:

$7\frac{5}{6}+7=14\frac{5}{6}$