Addition of Fractions: In combination with other operations

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

Therefore, we'll start by simplifying the expressions in parentheses first:
$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{2}{4}\cdot10:7:5 = \\ \frac{1}{2}\cdot10:7:5$

We calculated the expression inside the parentheses by adding the fractions, which we did by creating one fraction using the common denominator (4) which in this case is the denominator in all fractions, so we only added/subtracted the numerators (according to the fraction sign), then we reduced the resulting fraction,

We'll continue and note that between multiplication and division operations there is no defined precedence for either operation, therefore we'll calculate the result of the expression obtained in the last step step by step from left to right (which is the regular order in arithmetic operations), meaning we'll first perform the multiplication operation, which is the first from the left, and then we'll perform the division operation that comes after it, and so on:

$\frac{1}{2}\cdot10:7:5 =\\ \frac{1\cdot10}{2}:7:5 =\\ \frac{10}{2}:7:5 =\\ 5:7:5 =\\ \frac{5}{7}:5$

In the first step, we performed the multiplication of the fraction by the whole number, remembering that multiplying by a fraction means multiplying by the fraction's numerator, then we simplified the resulting fraction and reduced it (effectively performing the division operation that results from it), in the final step we wrote the division operation as a simple fraction, since this division operation yields a non-whole result,

We'll continue and to perform the final division operation, we'll remember that dividing by a number is the same as multiplying by its reciprocal, and therefore we'll replace the division operation with multiplication by the reciprocal:

$\frac{5}{7}:5 =\\ \frac{5}{7}\cdot\frac{1}{5}$

In this case we preferred to multiply by the reciprocal because the divisor in the expression is a fraction and it's more convenient to perform multiplication between fractions,

We will then perform the multiplication between the fractions we got in the last step, while remembering that multiplication between fractions is performed by multiplying numerator by numerator and denominator by denominator while maintaining the fraction line, then we'll simplify the resulting expression by reducing it:

$\frac{5}{7}\cdot\frac{1}{5} =\\ \frac{5\cdot1}{7\cdot5}=\\ \frac{5}{35}=\\ \frac{1}{7}$

Let's summarize the solution steps, we got that:

$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{1}{2}\cdot10:7:5 =\\ 5:7:5 =\\ \frac{5}{7}\cdot\frac{1}{5} =\\ \frac{1}{7}$

Therefore the correct answer is answer B.

To solve the expression $\frac{1}{2} - \frac{2}{8} + \frac{1}{4}$, we must first find a common denominator for the fractions involved.

Step 1: Identify a common denominator. The denominators are 2, 8, and 4. The smallest common multiple of these numbers is 8.

Step 2: Convert each fraction to have the common denominator of 8.

• The fraction $\frac{1}{2}$ can be written as $\frac{4}{8}$ because $1 \times 4 = 4$ and $2 \times 4 = 8$.
• The fraction $\frac{2}{8}$ is already expressed with 8 as the denominator.
• The fraction $\frac{1}{4}$ can be written as $\frac{2}{8}$ because $1 \times 2 = 2$ and $4 \times 2 = 8$.

Step 3: Substitute these equivalent fractions back into the original expression:

$\frac{4}{8} - \frac{2}{8} + \frac{2}{8}$

Step 4: Perform the subtraction and addition following the order of operations:

• Subtract: $\frac{4}{8} - \frac{2}{8} = \frac{2}{8}$
• Add: $\frac{2}{8} + \frac{2}{8} = \frac{4}{8}$

Step 5: Simplify the result:

$\frac{4}{8}$ simplifies to $\frac{1}{2}$ by dividing the numerator and denominator by 4.

Therefore, the value of the expression is $\frac{1}{2}$.

Let's try to find the lowest common denominator between 3, 15, and 5

To find the lowest common denominator, we need to find a number that is divisible by 3, 15, and 5

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 5

We'll multiply the second fraction by 1

We'll multiply the third fraction by 3

$\frac{2\times5}{3\times5}+\frac{2\times1}{15\times1}-\frac{4\times3}{5\times3}=\frac{10}{15}+\frac{2}{15}-\frac{12}{15}$

Now we'll add and then subtract:

$\frac{10+2-12}{15}=\frac{12-12}{15}=\frac{0}{15}$

We'll divide both the numerator and denominator by 0 and get:

$\frac{0}{15}=0$

Let's try to find the lowest common denominator between 3, 15, and 5

To find the lowest common denominator, we need to find a number that is divisible by 3, 15, and 5

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 5

We'll multiply the second fraction by 1

We'll multiply the third fraction by 3

$\frac{1\times5}{3\times5}+\frac{7\times1}{15\times1}-\frac{2\times3}{5\times3}=\frac{5}{15}+\frac{7}{15}-\frac{6}{15}$

Now we'll add and then subtract:

$\frac{5+7-6}{15}=\frac{12-6}{15}=\frac{6}{15}$

We'll divide both numerator and denominator by 3 and get:

$\frac{6:3}{15:3}=\frac{2}{5}$

To solve this problem, we'll perform the following steps:

• Step 1: Determine a common denominator for the fractions.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Perform the addition and subtraction operations on these fractions.
• Step 4: Simplify the resulting fraction, if possible.

Now, let's work through each step:

Step 1: To combine $\frac{3}{5}$, $\frac{1}{5}$, and $\frac{3}{15}$, identify the least common denominator (LCD). The denominators here are 5, 5, and 15. The least common multiple of 5 and 15 is 15. Therefore, our common denominator is 15.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 15:
$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$,
$\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$,
$\frac{3}{15}$ is already with the common denominator.

Step 3: Add and subtract the fractions:
$\frac{9}{15} + \frac{3}{15} = \frac{12}{15}$
$\frac{12}{15} - \frac{3}{15} = \frac{9}{15}$.

Step 4: Simplify the resulting fraction:
$\frac{9}{15} = \frac{3}{5}$ (dividing the numerator and denominator by their greatest common divisor, which is 3).

Therefore, the solution to the problem is $\frac{3}{5}$.

To solve this problem, we will follow these steps:

• Step 1: Find the least common denominator (LCD) for the fractions.
• Step 2: Convert each fraction to an equivalent fraction with the common denominator.
• Step 3: Perform the operations in the given order, subtraction first, then addition.

Now, let's work through each step:

Step 1: The denominators of the given fractions are 5 and 15. The least common multiple (LCM) of these numbers is 15, so 15 will be our common denominator.

Step 2: Convert each fraction to have the denominator of 15:
- $\frac{3}{5}$ is converted by multiplying both the numerator and denominator by 3, resulting in $\frac{9}{15}$.
- $\frac{1}{5}$ is converted by multiplying both the numerator and denominator by 3, yielding $\frac{3}{15}$.
- $\frac{3}{15}$ is already in terms of the common denominator.

Step 3: Perform the subtraction and addition:
- Start by subtracting $\frac{3}{15}$ from $\frac{9}{15}$:

$\frac{9}{15} - \frac{3}{15} = \frac{6}{15}$

Now, add $\frac{6}{15}$ and $\frac{3}{15}$:

$\frac{6}{15} + \frac{3}{15} = \frac{9}{15}$

Finally, simplify $\frac{9}{15}$ by dividing the numerator and denominator by their greatest common divisor, which is 3:

$\frac{9}{15} = \frac{3}{5}$

Therefore, the solution to the problem is $\frac{3}{5}$.

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

• Simplify each fraction.

• Identify the least common denominator (LCD).

• Convert each fraction to have this common denominator.

• Perform the addition and subtraction.

• Simplify the final result.

Let's work through each step:
Step 1: Simplify each fraction.
- $\frac{3}{6}$ simplifies to $\frac{1}{2}$ because both the numerator and denominator are divisible by 3.
- $\frac{2}{4}$ simplifies to $\frac{1}{2}$ because both the numerator and denominator are divisible by 2.
- $\frac{1}{12}$ is already in its simplest form.

Step 2: Identify the least common denominator (LCD).
- The denominators now are 2, 2, and 12. The LCD of 2 and 12 is 12.

Step 3: Convert each fraction to have this common denominator.
- $\frac{1}{2} = \frac{6}{12}$ (since $1 \times 6 = 6$ and $2 \times 6 = 12$)
- $\frac{1}{2} = \frac{6}{12}$ (similarly converted)
- $\frac{1}{12} = \frac{1}{12}$ (already has the denominator 12)

Step 4: Perform the addition and subtraction:
$\frac{6}{12} - \frac{6}{12} + \frac{1}{12} = \frac{6 - 6 + 1}{12} = \frac{1}{12}$

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

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

To solve the problem, follow these steps:

• Step 1: Convert each fraction to have a common denominator.
• Step 2: Add and subtract the fractions.
• Step 3: Simplify the result.

Let's work through these steps:

Step 1: Find the Least Common Denominator (LCD) of the fractions involved. The denominators are 6, 4, and 12. The LCM of these numbers is 12, so the LCD is 12.

Convert each fraction to this common denominator:

• $\frac{3}{6}$ becomes $\frac{3 \times 2}{6 \times 2} = \frac{6}{12}$
• $\frac{2}{4}$ becomes $\frac{2 \times 3}{4 \times 3} = \frac{6}{12}$
• $\frac{1}{12} \text{ remains } \frac{1}{12}$

Step 2: Perform the operations using these equivalent fractions: $\frac{6}{12} + \frac{6}{12} - \frac{1}{12} = \frac{6 + 6 - 1}{12} = \frac{11}{12}$

Step 3: Check if the result can be simplified further. In this case, $\frac{11}{12}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{11}{12}$.

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

• Simplify fractions if possible.
• Convert fractions to have a common denominator.
• Perform the operations in order, while considering appropriate signs.

Let's proceed step-by-step:

First, notice that $\frac{5}{10}$ simplifies to $\frac{1}{2}$, as both the numerator and denominator can be divided by 5.
Scheme now becomes:

Convert all fractions to have a denominator of 4:

• $\frac{1}{4}$ remains $\frac{1}{4}$.
• $\frac{3}{4}$ remains $\frac{3}{4}$.
• $\frac{1}{2}$ becomes $\frac{2}{4}$.

Our new expression is:

Perform the operations in left to right:

$\frac{1}{4} - \frac{3}{4} = -\frac{2}{4}$ which simplifies to $-\frac{1}{2}$

Now, combine with the next term:

$-\frac{1}{2} + \frac{2}{4} = 0$

Lastly, add the remaining term:

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

Thus, the correct answer is $\frac{1}{2}$, and it corresponds to choice 3.

To solve the expression $\frac{6}{7} - \frac{1}{2} + \frac{3}{14}$, we will follow these steps:

• Step 1: Find a common denominator for the fractions.
• Step 2: Convert each fraction to the common denominator.
• Step 3: Perform the subtraction and addition as required.
• Step 4: Simplify the result, if possible.

Let's work through the steps:

Step 1: The denominators are 7, 2, and 14. The least common multiple (LCM) of these numbers is 14.

Step 2: Convert each fraction:

• $\frac{6}{7}$ becomes $\frac{6 \times 2}{7 \times 2} = \frac{12}{14}$.
• $\frac{1}{2}$ becomes $\frac{1 \times 7}{2 \times 7} = \frac{7}{14}$.
• $\frac{3}{14}$ is already in the correct form as $\frac{3}{14}$.

Step 3: Perform the operations:

• Subtract: $\frac{12}{14} - \frac{7}{14} = \frac{5}{14}$.
• Add: $\frac{5}{14} + \frac{3}{14} = \frac{8}{14}$.

Step 4: Simplify the fraction if possible. Here, $\frac{8}{14}$ simplifies to $\frac{4}{7}$; however, since the given choices list $\frac{8}{14}$ and it matches, there is no need for further simplification within the context of this question.

Therefore, the solution to the problem is $\frac{8}{14}$.

To solve the expression $\frac{2}{7}+\frac{1}{2}-\frac{7}{14}$, follow these steps:

• Step 1: Find the least common multiple (LCM) of the denominators 7, 2, and 14. The LCM is 14.
• Step 2: Convert each fraction to have the denominator of 14:
      $\frac{2}{7}$ becomes $\frac{2 \times 2}{7 \times 2} = \frac{4}{14}$
      $\frac{1}{2}$ becomes $\frac{1 \times 7}{2 \times 7} = \frac{7}{14}$
      $\frac{7}{14}$ remains $\frac{7}{14}$ since it's already over 14.
• Step 3: Perform the operations in the expression $\frac{4}{14} + \frac{7}{14} - \frac{7}{14}$:
      First, add $\frac{4}{14} + \frac{7}{14} = \frac{11}{14}$.
      Then, subtract $\frac{11}{14} - \frac{7}{14} = \frac{4}{14}$.
• Step 4: The result is already simplified. Thus, the solution to the problem is $\frac{4}{14}$.

Therefore, the solution to the problem is $\mathbf{\frac{4}{14}}$, which corresponds to choice $\mathbf{3}$.

To solve the problem $\frac{2}{5} + \frac{1}{2} - \frac{1}{3}$, we will follow these steps:

• Step 1: Find the least common denominator (LCD) for $\frac{2}{5}$, $\frac{1}{2}$, and $\frac{1}{3}$.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Perform the arithmetic operations.
• Step 4: Simplify the result if necessary.

Now, let's proceed with the solution:
Step 1: The denominators are 5, 2, and 3. The least common multiple of these numbers is 30. Thus, the LCD is 30.

Step 2: Convert each fraction to have the common denominator of 30:
- Convert $\frac{2}{5}$ to a fraction with denominator 30: $\frac{2}{5} = \frac{2 \times 6}{5 \times 6} = \frac{12}{30}$.
- Convert $\frac{1}{2}$ to a fraction with denominator 30: $\frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30}$.
- Convert $\frac{1}{3}$ to a fraction with denominator 30: $\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}$.

Step 3: With all fractions having the same denominator, perform the operations:
$\frac{12}{30} + \frac{15}{30} - \frac{10}{30} = \frac{12 + 15 - 10}{30} = \frac{17}{30}$.

Step 4: Since $\frac{17}{30}$ is in its simplest form, no further simplification is needed.

Therefore, the correct answer is $\frac{17}{30}$.

To solve the expression $\frac{9}{10} - \frac{4}{5} + \frac{1}{2}$, we first seek a common denominator for the fractions. The denominators are 10, 5, and 2.

The least common multiple of these numbers is 10, as it is the smallest number that all denominators divide perfectly.

• Convert $\frac{4}{5}$ to a fraction with a denominator of 10: $\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$.
• Convert $\frac{1}{2}$ to a fraction with a denominator of 10: $\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

Now, the expression becomes $\frac{9}{10} - \frac{8}{10} + \frac{5}{10}$.

Perform the operations:

• Subtract: $\frac{9}{10} - \frac{8}{10} = \frac{1}{10}$.
• Add: $\frac{1}{10} + \frac{5}{10} = \frac{6}{10}$.

Thus, the value of the expression is $\frac{6}{10}$.

The correct answer is $\frac{6}{10}$.

To solve the expression $\frac{4}{10} + \frac{1}{5} - \frac{1}{2}$, we will follow these steps:

• Step 1: Find a Common Denominator
  The denominators we have are 10, 5, and 2. The least common denominator (LCD) among these numbers is 10.

• Step 2: Convert Fractions to Equivalent Fractions with the LCD
  - $\frac{4}{10}$ is already using 10 as the denominator.
  - $\frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10}$.
  - $\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

• Step 3: Perform the Arithmetic Operations
  Substitute the converted fractions into the original expression:
  $\frac{4}{10} + \frac{2}{10} - \frac{5}{10}$
  Combine the numerators over the common denominator:
  $\frac{4 + 2 - 5}{10} = \frac{1}{10}$

• Step 4: Simplify the Result
  The fraction $\frac{1}{10}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{1}{10}$.

To solve the problem, let's work through the following steps:

• Step 1: Identify the least common denominator (LCD) for all fractions.
  - The denominators are 8, 2, and 4. The LCM of these numbers is 8.

• Step 2: Convert each fraction to have this common denominator of 8.
  - $\frac{3}{8}$ is already with a denominator of 8.
  - $\frac{1}{2}$ can be rewritten as $\frac{4}{8}$ because $\frac{1 \times 4}{2 \times 4} = \frac{4}{8}$.
  - $\frac{1}{4}$ can be rewritten as $\frac{2}{8}$ because $\frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.

• Step 3: Perform the arithmetic operations.
  - Add $\frac{3}{8}$ and $\frac{4}{8}$, which gives $\frac{3 + 4}{8} = \frac{7}{8}$.
  - Subtract $\frac{2}{8}$ from $\frac{7}{8}$, giving $\frac{7 - 2}{8} = \frac{5}{8}$.

• Step 4: Simplify the answer if necessary.
  $\frac{5}{8}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{5}{8}$.

To solve the problem, we will find the result of $\frac{1}{4}+\frac{3}{4}+\frac{3}{20}-\frac{5}{10}$ by following these steps:

• Step 1: Combine fractions with the same denominator.
• Step 2: Find a common denominator for the remaining fractions.
• Step 3: Perform addition and subtraction accordingly.

Now, let's work through each step:

Step 1: Combine $\frac{1}{4}$ and $\frac{3}{4}$:
Both fractions have the denominator of 4, so we add their numerators:
$\frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1$.

Step 2: Deal with the remaining fractions $\frac{3}{20}$ and $\frac{1}{2}$:
We can change $\frac{5}{10}$ to $\frac{1}{2}$ and find a common denominator of 20 for $\frac{1}{2}$ and $\frac{3}{20}$:
Convert $\frac{1}{2}$ to $\frac{10}{20}$.

Step 3: Now the expression is $1 + \frac{3}{20} - \frac{10}{20}$. Simplify the fraction part:
Subtract $\frac{10}{20}$ from $\frac{3}{20}$:
$\frac{3}{20} - \frac{10}{20} = \frac{-7}{20}$.

So the expression becomes $1 + \frac{-7}{20}$. Write 1 as a fraction: $\frac{20}{20}$.

Step 4: Now add $\frac{20}{20} + \frac{-7}{20}$:
$\frac{20}{20} + \frac{-7}{20} = \frac{13}{20}$.

Therefore, the solution to the problem is $\frac{13}{20}$. The correct answer choice is ($\frac{13}{20}$).

To solve this problem, we need to first find a common denominator for all the fractions involved: $\frac{1}{4} - \frac{3}{4} + \frac{1}{2} - \frac{5}{10}$.

• Step 1: Identify the least common denominator (LCD). Here, the denominators are $4$, $4$, $2$, and $10$. The smallest number all these can divide into is $20$.
• Step 2: Convert each fraction to have this common denominator.
• $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$
• $\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$
• $\frac{1}{2} = \frac{1 \times 10}{2 \times 10} = \frac{10}{20}$
• $\frac{5}{10} = \frac{5 \times 2}{10 \times 2} = \frac{10}{20}$
• Step 3: Substitute these equivalent fractions back into the expression and simplify:
  $\frac{5}{20} - \frac{15}{20} + \frac{10}{20} - \frac{10}{20}.$
• Step 4: Perform the arithmetic operations:
• $\left(\frac{5}{20} - \frac{15}{20}\right) = -\frac{10}{20}.$
• $\left(-\frac{10}{20} + \frac{10}{20}\right) = 0.$
• $0 - \frac{10}{20} = -\frac{10}{20} = -\frac{1}{2}.$

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