All Operations in Mixed Fractions: More than two fractions

According to the order of operations, we must solve the exercise from left to right.

Let's note that in the first addition exercise, we have an addition between two halves that will give us a whole number, therefore:

$\frac{1}{2}+3\frac{1}{2}=4$

Now we will get the following exercise:

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

Let's note that we can simplify the mixed fraction:

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

Now the exercise we get is:

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

To solve this problem, we begin by converting each mixed number to an improper fraction:

• $2\frac{1}{2}$: Convert using the formula $\frac{ac + b}{c}$, where $a = 2$, $b = 1$, and $c = 2$. The improper fraction is $\frac{2 \times 2 + 1}{2} = \frac{5}{2}$.
• $3\frac{2}{4}$: Using the same formula, where $a = 3$, $b = 2$, and $c = 4$. The improper fraction is $\frac{3 \times 4 + 2}{4} = \frac{14}{4}$, which simplifies to $\frac{7}{2}$.
• $1\frac{3}{6}$: Again, using the formula, where $a = 1$, $b = 3$, and $c = 6$. The improper fraction is $\frac{1 \times 6 + 3}{6} = \frac{9}{6}$, which simplifies to $\frac{3}{2}$.

Next, we perform the addition and subtraction:

$\frac{5}{2} + \frac{7}{2} - \frac{3}{2}$

All fractions have the common denominator 2, so we can directly add and subtract the numerators:

$(5 + 7 - 3) / 2 = \frac{9}{2}$

Finally, convert the improper fraction $\frac{9}{2}$ back to a mixed number:

$9 \div 2 = 4$ with a remainder of 1, so $\frac{9}{2} = 4\frac{1}{2}$.

Therefore, the final solution to the problem is $4\frac{1}{2}$.

To solve the problem, we need to first convert the mixed numbers $2\frac{3}{4}$ and $3\frac{1}{2}$ to improper fractions and then perform the operations.

Step 1: Convert the mixed numbers to improper fractions.

• $2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$
• $3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}$

Step 2: Find a common denominator for all fractions involved, which is $4$.

• $\frac{11}{4}$ is already expressed with the denominator $4$.
• $\frac{1}{4}$ is already expressed with the denominator $4$.
• $\frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4}$

Step 3: Perform the operations.

• $\frac{11}{4} - \frac{1}{4} + \frac{14}{4}$
• Perform subtraction: $\frac{11}{4} - \frac{1}{4} = \frac{11 - 1}{4} = \frac{10}{4}$.
• Perform addition: $\frac{10}{4} + \frac{14}{4} = \frac{10 + 14}{4} = \frac{24}{4}$.

Step 4: Simplify the resulting fraction.

• $\frac{24}{4} = 6$

The final result of $2\frac{3}{4} - \frac{1}{4} + 3\frac{1}{2} = 6$.

Therefore, the solution to the problem is $6$.

To solve this problem of adding mixed numbers, follow these well-defined steps:

• Step 1: Convert the mixed numbers to improper fractions.
• Step 2: Find the least common denominator (LCD).
• Step 3: Convert each fraction to have the LCD as the denominator.
• Step 4: Sum the fractions, and finally, convert back to a mixed number if necessary.

Let us apply these steps to the given numbers:

Step 1: Convert to improper fractions:
- $4\frac{2}{5} = \frac{22}{5}$
- $1\frac{3}{10} = \frac{13}{10}$
- $2\frac{1}{20} = \frac{41}{20}$
- $2\frac{3}{5} = \frac{13}{5}$

Step 2: Find the least common denominator: The denominators are 5, 10, and 20. The LCD of these is 20.

Step 3: Convert fractions to have this LCD:
- $\frac{22}{5} = \frac{88}{20}$
- $\frac{13}{10} = \frac{26}{20}$
- $\frac{41}{20} \text{ remains } \frac{41}{20}$
- $\frac{13}{5} = \frac{52}{20}$

Step 4: Add the fractions:
$\frac{88}{20} + \frac{26}{20} + \frac{41}{20} + \frac{52}{20} = \frac{207}{20}$

Convert $\frac{207}{20}$ into a mixed number:
$207 \div 20 = 10$ remainder $7$, so we have $10\frac{7}{20}$.

Therefore, the sum of the mixed numbers $4\frac{2}{5} + 1\frac{3}{10} + 2\frac{1}{20} + 2\frac{3}{5}$ is $10\frac{7}{20}$.

To solve this problem, we will perform the following steps:

• Convert each mixed fraction into an improper fraction.
• Find a common denominator for these fractions.
• Perform the subtraction and addition operations.
• Convert the result back into a mixed number.
• Simplify the final answer if necessary.

Let’s proceed with these steps.

Step 1: Convert to improper fractions
For $5\frac{1}{2}$:
$5\frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{11}{2}$

For $2\frac{3}{4}$:
$2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$

For $4\frac{2}{6}$ (simplified as $4\frac{1}{3}$):
$4\frac{2}{6} = 4\frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3}$

Step 2: Find a common denominator
The denominators are $2$, $4$, and $3$. The least common multiple of these is $12$.
Convert each fraction:
$\frac{11}{2} = \frac{11 \times 6}{2 \times 6} = \frac{66}{12}$
$\frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{33}{12}$
$\frac{13}{3} = \frac{13 \times 4}{3 \times 4} = \frac{52}{12}$

Step 3: Perform the operations
First, perform the subtraction:
$\frac{66}{12} - \frac{33}{12} = \frac{33}{12}$

Now, add $\frac{33}{12}$ and $\frac{52}{12}$:
$\frac{33}{12} + \frac{52}{12} = \frac{85}{12}$

Step 4: Convert back to a mixed number
$\frac{85}{12}$ as a mixed number is $7\frac{1}{12}$ because:
$\frac{85}{12} = 7$ remainder $1$, thus $7\frac{1}{12}$.

Step 5: Simplify
The fraction $\frac{1}{12}$ is already in its simplest form.

Therefore, the correct answer is $7\frac{1}{12}$.

To solve this problem, we'll convert each mixed number to an improper fraction and then find a common denominator for the addition:

• Convert each mixed number to an improper fraction:
• $6\frac{2}{4} = \frac{6 \times 4 + 2}{4} = \frac{26}{4}$
• $1\frac{4}{6} = \frac{1 \times 6 + 4}{6} = \frac{10}{6}$
• $2\frac{7}{12} = \frac{2 \times 12 + 7}{12} = \frac{31}{12}$
• Find the least common denominator (LCD) of 4, 6, and 12, which is 12.
• Rewrite each fraction with the common denominator:
• $\frac{26}{4} = \frac{26 \times 3}{4 \times 3} = \frac{78}{12}$
• $\frac{10}{6} = \frac{10 \times 2}{6 \times 2} = \frac{20}{12}$
• $\frac{31}{12}$ is already over 12.
• Add the fractions:
• $\frac{78}{12} + \frac{20}{12} + \frac{31}{12} = \frac{78 + 20 + 31}{12} = \frac{129}{12}$
• Convert $\frac{129}{12}$ to a mixed number:
• Divide 129 by 12. This yields 10 with a remainder of 9.
• Therefore, the mixed number form is $10\frac{9}{12}$.
• Simplify $\frac{9}{12}$ to $\frac{3}{4}$.

Thus, the final result of the addition is $10\frac{3}{4}$.

The correct answer matches choice 4: $10\frac{3}{4}$.

To solve this problem, we'll proceed with these steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Calculate the least common denominator of the fractions.
• Step 3: Express each fraction with this common denominator.
• Step 4: Add the fractions together.
• Step 5: Simplify the result and convert back to a mixed number if necessary.

Let's begin the solution:

Step 1: Convert the mixed numbers to improper fractions:

• $6\frac{2}{7} = \frac{6 \times 7 + 2}{7} = \frac{44}{7}$
• $1\frac{3}{14} = \frac{1 \times 14 + 3}{14} = \frac{17}{14}$
• $2\frac{3}{7} = \frac{2 \times 7 + 3}{7} = \frac{17}{7}$
• $1\frac{1}{14} = \frac{1 \times 14 + 1}{14} = \frac{15}{14}$

Step 2: Determine the least common denominator (LCD). Here, the denominators are 7 and 14. The LCD of 7 and 14 is 14.

Step 3: Express each fraction with the common denominator of 14:

• $\frac{44}{7} = \frac{44 \times 2}{14} = \frac{88}{14}$
• $\frac{17}{14} = \frac{17}{14}$ (already with the denominator 14)
• $\frac{17}{7} = \frac{17 \times 2}{14} = \frac{34}{14}$
• $\frac{15}{14} = \frac{15}{14}$ (already with the denominator 14)

Step 4: Add the fractions together:

$\frac{88}{14} + \frac{17}{14} + \frac{34}{14} + \frac{15}{14} = \frac{154}{14}$

Step 5: Simplify the result:

$\frac{154}{14} = \frac{11 \times 14}{14} = 11$

Thus, the solution to the problem is $11$.

To solve this problem, we will follow these steps:

• Step 1: Convert each mixed number into an improper fraction.
• Step 2: Identify the least common denominator (LCD) for the fractions.
• Step 3: Add the fractions together.
• Step 4: Convert the resulting improper fraction back to a mixed number.

Now, let's work through each step in detail:

Step 1: Convert each mixed number into an improper fraction.

• $6\frac{2}{9} = \frac{(6 \times 9) + 2}{9} = \frac{54 + 2}{9} = \frac{56}{9}$
• $1\frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}$
• $1\frac{5}{9} = \frac{(1 \times 9) + 5}{9} = \frac{9 + 5}{9} = \frac{14}{9}$

Step 2: Identify the least common denominator (LCD) for the fractions.

The denominators are 9 and 3. The least common multiple of these is 9.

Step 3: Add the fractions.

• First, convert all fractions to have the same denominator.
• $\frac{5}{3}$ should be converted: $\frac{5 \times 3}{3 \times 3} = \frac{15}{9}$.
• Add: $\frac{56}{9} + \frac{15}{9} + \frac{14}{9} = \frac{85}{9}$.

Step 4: Convert $\frac{85}{9}$ back to a mixed number.

• Perform the division: $85 ÷ 9 = 9$ remainder $4$.
• Therefore, $\frac{85}{9} = 9\frac{4}{9}$.

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

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:

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

To solve the given problem $10\frac{2}{7} - 2\frac{3}{7} + 7\frac{1}{6}$, we'll proceed with the following steps:

• Step 1: Convert the mixed numbers to improper fractions.
  For $10\frac{2}{7}$, convert to $\frac{72}{7}$, since $10 \times 7 + 2 = 70 + 2 = 72$.
  For $2\frac{3}{7}$, convert to $\frac{17}{7}$, since $2 \times 7 + 3 = 14 + 3 = 17$.
  For $7\frac{1}{6}$, convert to $\frac{43}{6}$, since $7 \times 6 + 1 = 42 + 1 = 43$.
• Step 2: Find a common denominator.
  The denominators are 7 and 6. The least common denominator (LCD) is 42.
• Step 3: Convert fractions to the common denominator.
  Convert $\frac{72}{7}$ to $\frac{432}{42}$ (since $72 \times 6 = 432$).
  Convert $\frac{17}{7}$ to $\frac{102}{42}$ (since $17 \times 6 = 102$).
  Convert $\frac{43}{6}$ to $\frac{301}{42}$ (since $43 \times 7 = 301$).
• Step 4: Perform the operations.
  First, subtract: $\frac{432}{42} - \frac{102}{42} = \frac{330}{42}$.
  Then, add: $\frac{330}{42} + \frac{301}{42} = \frac{631}{42}$.
• Step 5: Simplify the result.
  Since 631 and 42 have no common factors other than 1, the fraction is already in its simplest form. Convert back to a mixed number: $631 \div 42 = 15$ remainder $1$, so $\frac{631}{42} = 15\frac{1}{42}$.

Therefore, the solution to the problem is $15\frac{1}{42}$.

According to the rules of the order of operations in arithmetic, we solve the exercise from left to right.

Let's note that:

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

We should obtain the following exercise:

$1+2\frac{3}{4}=3\frac{3}{4}$

Let's solve the exercise from left to right.

We will combine the left expression in the following way:

$\frac{6+8}{7}x=\frac{14}{7}x=2x$

Now we get:

$2x+3\frac{2}{3}x=5\frac{2}{3}x$