All Operations in Fractions: In combination with other operations

To solve the problem $\frac{1}{4} : \frac{1}{2} + \frac{1}{4}$, follow these steps:

Step 1: Perform the division $\frac{1}{4} : \frac{1}{2}$.

• When dividing by a fraction, multiply by the reciprocal. Hence, $\frac{1}{4} : \frac{1}{2}$ becomes $\frac{1}{4} \times \frac{2}{1}$.
• Multiply the numerators: $1 \times 2 = 2$.
• Multiply the denominators: $4 \times 1 = 4$.
• So, $\frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2}$ after simplification.

Step 2: Now add the result from Step 1 to $\frac{1}{4}$.

• We need to add $\frac{1}{2} + \frac{1}{4}$.
• Convert $\frac{1}{2}$ to have a common denominator with $\frac{1}{4}$. $\frac{1}{2} = \frac{2}{4}$.
• Add the fractions: $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.

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

To solve this fraction problem, follow these steps:

• Step 1: Evaluate the Division:
  The expression starts with $\frac{2}{3} \div \frac{3}{4}$. To divide fractions, multiply by the reciprocal:
  $\frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9}$
• Step 2: Add Fractions:
  Now add $\frac{8}{9}$ to $\frac{1}{9}$. Since the denominators are the same, add the numerators directly:
  $\frac{8}{9} + \frac{1}{9} = \frac{8 + 1}{9} = \frac{9}{9} = 1$

Thus, the solution to the expression $\frac{2}{3}:\frac{3}{4} + \frac{1}{9}$ is $1$.

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

• Step 1: First, interpret the division $\frac{1}{2} : \frac{1}{2}$ as multiplication by the reciprocal. This becomes $\frac{1}{2} \times \frac{2}{1}$.
• Step 2: Perform the multiplication: $\frac{1}{2} \times \frac{2}{1} = \frac{1 \times 2}{2 \times 1} = \frac{2}{2} = 1.$
• Step 3: Next, subtract $\frac{1}{4}$ from 1: $1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}.$

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

To solve the problem, we'll follow the outlined steps:

• Step 1: Calculate the division $\frac{3}{5} : \frac{5}{6}$.
• Step 2: Add $\frac{1}{5}$ to the result from Step 1.
• Step 3: Simplify the resulting fraction.

Now, let's work through each step:

Step 1: To divide $\frac{3}{5}$ by $\frac{5}{6}$, we multiply by the reciprocal of $\frac{5}{6}$. This gives us:

$\frac{3}{5} \times \frac{6}{5} = \frac{3 \times 6}{5 \times 5} = \frac{18}{25}$

Step 2: Now, add $\frac{1}{5}$ to $\frac{18}{25}$. First, we convert $\frac{1}{5}$ to the same denominator as $\frac{18}{25}$:

$\frac{1}{5} = \frac{5}{25}$

Step 3: Add $\frac{18}{25}$ and $\frac{5}{25}$:

$\frac{18}{25} + \frac{5}{25} = \frac{18 + 5}{25} = \frac{23}{25}$

Thus, the solution to the problem is $\frac{23}{25}$.

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

• Step 1: Calculate the division $\frac{3}{4} : \frac{5}{4}$.
• Step 2: Use the formula $\frac{a}{b} : \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.
• Step 3: Simplify the division result.
• Step 4: Add $\frac{1}{2}$ to the result of the division.
• Step 5: Simplify the addition result to get the final answer.

Now, let's work through each step:

Step 1: We need to calculate $\frac{3}{4} : \frac{5}{4}$. Using the division of fractions formula, this becomes:

$\frac{3}{4} \times \frac{4}{5} = \frac{3 \times 4}{4 \times 5} = \frac{12}{20}$.

Step 2: Simplify $\frac{12}{20}$. Divide the numerator and the denominator by their greatest common divisor, which is 4:

$\frac{12 \div 4}{20 \div 4} = \frac{3}{5}$.

Step 3: Add $\frac{1}{2}$ to the result $\frac{3}{5}$:

The common denominator for addition is 10. Therefore:

$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$ and $\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

Add these two fractions:

$\frac{6}{10} + \frac{5}{10} = \frac{6 + 5}{10} = \frac{11}{10}$.

Therefore, the solution to the problem is $\frac{11}{10}$.

To solve this mathematical problem, follow these steps:

• Step 1: Multiply the fractions $\frac{3}{4}$ and $\frac{3}{4}$.
• Step 2: Subtract $\frac{1}{4}$ from the product obtained in Step 1.

Let's execute each step in detail:

Step 1: Calculate the product of $\frac{3}{4}$ and $\frac{3}{4}$.

To multiply two fractions, multiply their numerators and their denominators separately:

$\frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16}$

Step 2: Subtract $\frac{1}{4}$ from $\frac{9}{16}$.

Before we subtract $\frac{1}{4}$ from $\frac{9}{16}$, we need a common denominator. The common denominator for these fractions is 16:

$\frac{1}{4} = \frac{1 \times 4}{4 \times 4} = \frac{4}{16}$

Now subtract $\frac{4}{16}$ from $\frac{9}{16}$:

$\frac{9}{16} - \frac{4}{16} = \frac{9 - 4}{16} = \frac{5}{16}$

Therefore, the solution to the given problem is $\frac{5}{16}$.

To solve $\frac{1}{2} \times \frac{1}{2} + \frac{3}{4}$, follow these steps:

• Step 1: Multiply $\frac{1}{2} \times \frac{1}{2}$ by using the multiplication of fractions formula:
  $\frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4}$.
• Step 2: Add $\frac{1}{4}$ to $\frac{3}{4}$.
  Since $\frac{1}{4}$ and $\frac{3}{4}$ already have the same denominator, the addition can be done directly:
  $\frac{1}{4} + \frac{3}{4} = \frac{1 + 3}{4} = \frac{4}{4} = 1$.

Therefore, the correct solution to the expression is $1$.

To solve the problem $\frac{3}{5} \times \frac{2}{3} + \frac{2}{5}$, we proceed with the following steps:

• Step 1: Multiply the fractions $\frac{3}{5}$ and $\frac{2}{3}$.

The multiplication yields:

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

• Step 2: Simplify the product $\frac{6}{15}$.

Both 6 and 15 share a common factor of 3:

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

• Step 3: Add $\frac{2}{5}$ to the simplified result $\frac{2}{5}$.

Since the fractions $\frac{2}{5}$ and $\frac{2}{5}$ have the same denominator, add the numerators while keeping the denominator:

$\frac{2}{5} + \frac{2}{5} = \frac{2+2}{5} = \frac{4}{5}$

Therefore, the solution to the problem is $\frac{4}{5}$.

To solve this problem, let's follow these steps:

• Step 1: Multiply the fractions. Calculate $\frac{2}{3} \times \frac{1}{3}$.
• Step 2: Add the product to another fraction. Add the result to $\frac{2}{9}$.

Now, let's work through the calculations:

Step 1: Multiply $\frac{2}{3}$ by $\frac{1}{3}$.

The formula for multiplying fractions is:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.

Substitute the values:

$\frac{2}{3} \times \frac{1}{3} = \frac{2 \times 1}{3 \times 3} = \frac{2}{9}$.

Step 2: Add $\frac{2}{9}$ to the product.

We found in Step 1 that $\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$.

Now add $\frac{2}{9} + \frac{2}{9} = \frac{2 + 2}{9} = \frac{4}{9}$.

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

To solve the problem $\frac{3}{4} \times \frac{1}{2} + \frac{5}{8}$, we'll follow these steps:

• Step 1: Multiply the fractions $\frac{3}{4} \times \frac{1}{2}$.
• Step 2: Add the result to $\frac{5}{8}$.

Now, let's work through the steps:

Step 1: Compute the product of the first two fractions:
$\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$

Step 2: Add the resulting fraction to $\frac{5}{8}$ by finding a common denominator:

The fractions $\frac{3}{8}$ and $\frac{5}{8}$ already have the same denominator, so we can simply add them:
$\frac{3}{8} + \frac{5}{8} = \frac{3 + 5}{8} = \frac{8}{8} = 1$

Therefore, the solution to the problem is $1$.

To solve the expression $\frac{4}{4} \times \frac{1}{2} + \frac{3}{8}$, follow these steps:

• Step 1: Simplify $\frac{4}{4}$. Since $\frac{4}{4} = 1$, the expression becomes $1 \times \frac{1}{2} + \frac{3}{8}$.
• Step 2: Perform the multiplication.
  Calculate $1 \times \frac{1}{2} = \frac{1}{2}$.
• Step 3: Add $\frac{1}{2}$ to $\frac{3}{8}$.
  First, convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8: $\frac{1}{2} = \frac{4}{8}$.
• Step 4: Now, add the fractions: $\frac{4}{8} + \frac{3}{8} = \frac{7}{8}$.

Thus, the final result is $\frac{7}{8}$.

To solve the given problem, we will follow these steps:

• Step 1: Perform the multiplication of the fractions.
• Step 2: Simplify the result, if applicable.
• Step 3: Add the simplified fractional result to the given fraction, ensuring the denominators align properly.
• Step 4: Simplify the final result, if necessary.

Let's go through each step:

Step 1: Multiply the fractions $\frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}$.

Step 2: The result from step 1 is $\frac{4}{9}$, which cannot be further simplified.

Step 3: Add the result from Step 2 to $\frac{4}{9}$ given in the problem:
We have two fractions $\frac{4}{9}$ and $\frac{4}{9}$, and since they already have a common denominator, we add them directly:
$\frac{4}{9} + \frac{4}{9} = \frac{4 + 4}{9} = \frac{8}{9}$.

Step 4: The fraction $\frac{8}{9}$ is already in its simplest form.

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

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

• Step 1: Multiply the given fractions.
• Step 2: Simplify if necessary.
• Step 3: Perform the addition of resulting fractions.

Now, let's work through each step:

Step 1: Multiply the fractions:
$\frac{1}{4} \times \frac{1}{2} = \frac{1 \times 1}{4 \times 2} = \frac{1}{8}$

Step 2: We find that the result is already simplified.

Step 3: Add $\frac{1}{8}$ to $\frac{3}{8}$:
$\frac{1}{8} + \frac{3}{8} = \frac{1 + 3}{8} = \frac{4}{8} = \frac{1}{2}$

The fractions have the same denominator, allowing for direct addition.

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

To solve this problem, we'll approach it in the following steps:

Step 1: Perform the Multiplication
The expression begins with multiplying two fractions: $\frac{1}{4} \times \frac{4}{5}$. Using the formula for multiplying fractions, we get:
$\frac{1 \times 4}{4 \times 5} = \frac{4}{20}$
Simplifying $\frac{4}{20}$ by dividing both numerator and denominator by 4 gives:
$\frac{1}{5}$

Step 2: Add the Result to the Second Fraction
Now, we need to add $\frac{1}{5}$ to $\frac{11}{20}$. To do this, we first find a common denominator.
The least common denominator between 5 and 20 is 20. Convert $\frac{1}{5}$ to twentieths:
$\frac{1}{5} = \frac{4}{20}$
Now add $\frac{4}{20}$ to $\frac{11}{20}$:
$\frac{4}{20} + \frac{11}{20} = \frac{15}{20}$

Step 3: Simplify the Final Result
Simplify $\frac{15}{20}$ by dividing the numerator and the denominator by 5:
$\frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

Therefore, the solution to the problem is $\frac{3}{4}$. This matches choice 1, which is $\frac{3}{4}$.

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

• Step 1: Multiply the first two fractions.
• Step 2: Add the result to the third fraction.
• Step 3: Simplify the final result.

Now, let's work through each step:
Step 1: Multiply $\frac{4}{5}$ by $\frac{1}{2}$. According to the multiplication rule for fractions, we have:
$\frac{4}{5} \times \frac{1}{2} = \frac{4 \times 1}{5 \times 2} = \frac{4}{10}$ Step 2: We need to add $\frac{4}{10}$ to $\frac{3}{10}$. Since these fractions have the same denominator, we can add them directly:
$\frac{4}{10} + \frac{3}{10} = \frac{4 + 3}{10} = \frac{7}{10}$ Step 3: The sum $\frac{7}{10}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{7}{10}$, which matches choice $\text{(3)}$.

To solve the exercise $\frac{1}{10} + \frac{3}{5} - \frac{1}{2}$, we must follow these steps:

Step 1: Find the Least Common Denominator (LCD).
The denominators we have are 10, 5, and 2. The LCD for these numbers is 10.

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

Step 3: Perform the addition and subtraction.
Now operate: $\frac{1}{10} + \frac{6}{10} - \frac{5}{10} = \frac{1 + 6 - 5}{10} = \frac{2}{10}$

Step 4: Simplify the result.
The fraction $\frac{2}{10}$ simplifies to $\frac{1}{5}$ because both the numerator and denominator are divisible by 2.

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

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

• Step 1: Determine the least common multiple (LCM) of the denominators $10$, $5$, and $2$.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Perform the subtraction and addition operations sequentially.

Now, let's work through each step:
Step 1: The denominators are $10$, $5$, and $2$. The LCM of these numbers is $10$.
Step 2: Convert each fraction:
- $\frac{11}{10}$ already has the denominator $10$.
- Convert $\frac{4}{5}$ to have a denominator of $10$:
$\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$.
- Convert $\frac{1}{2}$ to have a denominator of $10$:
$\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.
Step 3: Perform the operations:
- First, subtract: $\frac{11}{10} - \frac{8}{10} = \frac{11 - 8}{10} = \frac{3}{10}$.
- Then, add: $\frac{3}{10} + \frac{5}{10} = \frac{3 + 5}{10} = \frac{8}{10} = \frac{4}{5}$ after simplifying.

Therefore, the solution to the problem is $\frac{4}{5}$.

To solve the expression $\frac{2}{7} + \frac{1}{2} - \frac{7}{14}$, we need to add and subtract fractions, which requires a common denominator.

• Step 1: Identify the denominators and find the least common denominator (LCD):
  • The denominators are 7, 2, and 14.
  • The LCM of 7, 2, and 14 is 14.
• Step 2: Convert each fraction to have the denominator 14:
  • $\frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14}$
  • $\frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}$
  • $\frac{7}{14}$ is already in the form with denominator 14.
• Step 3: Perform the operations using the common denominator:
  • Add the first two fractions: $\frac{4}{14} + \frac{7}{14} = \frac{11}{14}$
  • Subtract the third fraction: $\frac{11}{14} - \frac{7}{14} = \frac{4}{14}$
• Step 4: Simplify the result if necessary:
  • $\frac{4}{14}$ is already simplified to its simplest form.

Therefore, the solution to the expression $\frac{2}{7} + \frac{1}{2} - \frac{7}{14}$ is $\frac{4}{14}$, which matches choice 3.

To solve the problem $\frac{5}{8} + \frac{1}{2} - \frac{1}{4}$, we will follow these steps:

Step 1: Find the least common denominator (LCD).
The denominators are 8, 2, and 4. The least common multiple of these numbers is 8.

Step 2: Convert each fraction to have a denominator of 8.
- $\frac{5}{8}$ already has the denominator 8.
- $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$.
- $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.

Step 3: Perform the arithmetic operations.
First, add $\frac{5}{8}$ and $\frac{4}{8}$:
$\frac{5}{8} + \frac{4}{8} = \frac{5 + 4}{8} = \frac{9}{8}$.
Then, subtract $\frac{2}{8}$ from $\frac{9}{8}$:
$\frac{9}{8} - \frac{2}{8} = \frac{9 - 2}{8} = \frac{7}{8}$.

Therefore, the solution to the problem is $\frac{7}{8}$.

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

• Step 1: Perform the multiplication operation:
  Calculate $\frac{3}{4} \cdot \frac{1}{2}$:
  Multiply the numerators: $3 \times 1 = 3$.
  Multiply the denominators: $4 \times 2 = 8$.
  Thus, $\frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8}$.
• Step 2: Perform the subtraction operation:
  Now, subtract $\frac{1}{4}$ from $\frac{3}{8}$.
  Before subtracting, we need a common denominator. The least common denominator of $8$ and $4$ is $8$.
  Convert $\frac{1}{4}$ to have a denominator of $8$:
  $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.
  Now, subtract the fractions:
  $\frac{3}{8} - \frac{2}{8} = \frac{3 - 2}{8} = \frac{1}{8}$.

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