All Operations in Fractions: Finding a Common Denominator by Multiplying the Denominators

To solve this problem, we'll begin by finding a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{4}$.
Step 1: Identify the denominators, which are 3 and 4. Multiply these to get a common denominator: $3 \times 4 = 12$.

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 12.

• To convert $\frac{1}{3}$ to a denominator of 12, multiply both the numerator and the denominator by 4: $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$.
• To convert $\frac{1}{4}$ to a denominator of 12, multiply both the numerator and the denominator by 3: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.

Step 3: Add the resulting fractions: $\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}$.

Thus, the sum of $\frac{1}{3}$ and $\frac{1}{4}$ is $\frac{7}{12}$.

To solve the problem of adding $\frac{1}{4}$ and $\frac{3}{6}$, we need to find their sum using a common denominator.

Step 1: Identify the Least Common Denominator (LCD)
The denominators of the fractions are 4 and 6. The LCM of 4 and 6, which will be the least common denominator, is 12.

Step 2: Convert each fraction to an equivalent fraction with the denominator of 12.
For $\frac{1}{4}$: Multiply the numerator and denominator by 3 to get $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.
For $\frac{3}{6}$: Multiply the numerator and denominator by 2 to get $\frac{3 \times 2}{6 \times 2} = \frac{6}{12}$.

Step 3: Add the fractions $\frac{3}{12} + \frac{6}{12} = \frac{3 + 6}{12} = \frac{9}{12}$.

Step 4: Simplify the resulting fraction if necessary.
In this case, $\frac{9}{12}$ can be simplified. The greatest common divisor of 9 and 12 is 3, so $\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4}$.

Therefore, the sum of $\frac{1}{4} + \frac{3}{6}$ is $\frac{3}{4}$, but in the context of the provided answer choices, we are looking for $\frac{9}{12}$ initially, which does match the simplified result before reducing.

The correct answer is therefore $\frac{9}{12}$, which corresponds to Choice 3.

To solve the given problem $\frac{1}{2} - \frac{2}{7}$, we need to follow these steps:

• Step 1: Find a common denominator for the fractions. The denominators are 2 and 7. The common denominator can be found by multiplying these two numbers: $2 \times 7 = 14$.
• Step 2: Convert the fractions to equivalent fractions with the common denominator. For $\frac{1}{2}$, multiply the numerator and the denominator by 7: $\frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}$. For $\frac{2}{7}$, multiply the numerator and the denominator by 2: $\frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14}$.
• Step 3: Subtract the fractions. With a common denominator, subtract the numerators: $\frac{7}{14} - \frac{4}{14} = \frac{7 - 4}{14} = \frac{3}{14}$.
• Step 4: Simplify the resulting fraction if needed. The fraction $\frac{3}{14}$ is already in its simplest form.

Thus, the solution to the problem is $\frac{3}{14}$.

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

First, we need to find a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{5}$. The denominators are 3 and 5, and their least common multiple (LCM) is 15.

We will convert each fraction to an equivalent fraction with the denominator 15:

• To convert $\frac{1}{3}$ to a fraction with denominator 15, multiply both the numerator and the denominator by 5: $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$
• To convert $\frac{1}{5}$ to a fraction with denominator 15, multiply both the numerator and the denominator by 3: $\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

Now that both fractions have the same denominator, we can subtract the numerators:

Therefore, the solution to the problem is $\frac{2}{15}$.

To solve the problem, let's follow a structured approach:

• Step 1: Determine the least common multiple (LCM) of the denominators (5 and 4). The LCM of 5 and 4 is 20.
• Step 2: Adjust each fraction to have the common denominator of 20.
  For $\frac{2}{5}$, multiply both numerator and denominator by 4 to get $\frac{8}{20}$.
  For $\frac{1}{4}$, multiply both numerator and denominator by 5 to get $\frac{5}{20}$.
• Step 3: Now, add the two fractions:
  $\frac{8}{20} + \frac{5}{20} = \frac{8 + 5}{20} = \frac{13}{20}$.
• Step 4: Verify if the fraction needs simplification. In this case, $\frac{13}{20}$ is already in its simplest form.

The resulting fraction after adding $\frac{2}{5}$ and $\frac{1}{4}$ is $\frac{13}{20}$.

To solve the problem of adding $\frac{3}{5}$ and $\frac{1}{3}$, the solution steps are as follows:

• Step 1: Identify a common denominator. Multiply the denominators: $5 \times 3 = 15$.
• Step 2: Convert each fraction to have this common denominator.
• Convert $\frac{3}{5}$: Multiply both numerator and denominator by 3: $\frac{3 \times 3}{5 \times 3} = \frac{9}{15}$.
• Convert $\frac{1}{3}$: Multiply both numerator and denominator by 5: $\frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Add the two fractions now that they have the same denominator: $\frac{9}{15} + \frac{5}{15} = \frac{9+5}{15} = \frac{14}{15}$.
• Step 4: Simplify if possible. In this case, $\frac{14}{15}$ is already in its simplest form.

Thus, the result of adding $\frac{3}{5}$ and $\frac{1}{3}$ is $\frac{14}{15}$, which corresponds to choice id "3" in the provided multiple-choice options.

To solve the problem of adding two fractions, follow these steps:

• Step 1: Identify the fractions involved: $\frac{1}{5}$ and $\frac{2}{3}$.
• Step 2: Find a common denominator. Multiply the denominators: $5 \times 3 = 15$.
• Step 3: Convert each fraction to have the common denominator of 15:
• Convert $\frac{1}{5}$ by multiplying both numerator and denominator by 3: $\frac{1}{5} \times \frac{3}{3} = \frac{3}{15}$
• Convert $\frac{2}{3}$ by multiplying both numerator and denominator by 5: $\frac{2}{3} \times \frac{5}{5} = \frac{10}{15}$
• Step 4: Add the converted fractions: $\frac{3}{15} + \frac{10}{15} = \frac{13}{15}$

Therefore, the sum of $\frac{1}{5} + \frac{2}{3}$ is $\frac{13}{15}$.

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

• Step 1: Identify a common denominator for the fractions.
• Step 2: Convert the fractions to equivalent fractions with the common denominator.
• Step 3: Add the fractions by summing the numerators.
• Step 4: Simplify the resulting fraction if necessary.

Now, let's work through each step:

Step 1: Identify a common denominator.
The denominators of the fractions are 6 and 3.
The least common multiple (LCM) of 6 and 3 is 6.

Step 2: Convert each fraction to equivalent fractions with a common denominator.
$\frac{5}{6}$ is already expressed with the denominator 6.
To convert $\frac{2}{3}$ to a fraction with the denominator 6, we multiply both the numerator and the denominator by 2:
$\frac{2}{3} \times \frac{2}{2} = \frac{4}{6}$.

Step 3: Add the fractions.
Now that both fractions have the same denominator, we can add them:
$\frac{5}{6} + \frac{4}{6} = \frac{9}{6}$.

Step 4: Simplify the resulting fraction.
The fraction $\frac{9}{6}$ can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 3:
$\frac{9}{6} = \frac{9 \div 3}{6 \div 3} = \frac{3}{2}$.

Therefore, the solution to the problem is $\frac{3}{2}$.

To solve the addition of fractions $\frac{1}{10} + \frac{1}{3}$, we must first find a common denominator.

• Step 1: Find the Least Common Multiple (LCM) of the denominators, 10 and 3. By multiplying these denominators, the LCM is $10 \times 3 = 30$.

• Step 2: Rewrite each fraction with the common denominator of 30:
  - Convert $\frac{1}{10}$ to an equivalent fraction with a denominator of 30. Multiply both numerator and denominator by 3: $\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}$
  - Convert $\frac{1}{3}$ to an equivalent fraction with a denominator of 30. Multiply both numerator and denominator by 10: $\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}$

• Step 3: Add the equivalent fractions: $\frac{3}{30} + \frac{10}{30} = \frac{3 + 10}{30} = \frac{13}{30}$

• Step 4: Simplify the resulting fraction. Since 13 is a prime number and does not divide 30, $\frac{13}{30}$ is already in its simplest form.

Thus, the sum of $\frac{1}{10}$ and $\frac{1}{3}$ is $\frac{13}{30}$.

The correct answer is $\frac{13}{30}$, which corresponds to choice 4.

To solve the problem of adding $\frac{1}{4}$ and $\frac{3}{6}$, we perform the following steps:

• Step 1: Find the least common multiple (LCM) of the denominators $4$ and $6$. The LCM of $4$ and $6$ is $12$.
• Step 2: Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of $12$.
  Multiply both the numerator and denominator of $\frac{1}{4}$ by $3$ to get $\frac{3}{12}$.
• Step 3: Convert $\frac{3}{6}$ to an equivalent fraction with a denominator of $12$.
  Multiply both the numerator and denominator of $\frac{3}{6}$ by $2$ to get $\frac{6}{12}$.
• Step 4: Add the equivalent fractions $\frac{3}{12} + \frac{6}{12}$.
• Step 5: Combine the numerators while keeping the common denominator: $\frac{3+6}{12} = \frac{9}{12}$.
• Step 6: Simplify $\frac{9}{12}$ by dividing the numerator and the denominator by their greatest common divisor, which is $3$, resulting in $\frac{3}{4}$.

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

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

• Step 1: Identify the common denominator.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Add the converted fractions.
• Step 4: Simplify the result.

Now, let's work through each step:
Step 1: The denominators are 7 and 8. Their product is $7 \times 8 = 56$. So, the common denominator is 56.
Step 2: Convert $\frac{1}{7}$ to have a denominator of 56 by multiplying numerator and denominator by 8: $\frac{1 \times 8}{7 \times 8} = \frac{8}{56}$.
Convert $\frac{1}{8}$ to have a denominator of 56 by multiplying numerator and denominator by 7: $\frac{1 \times 7}{8 \times 7} = \frac{7}{56}$.
Step 3: Add these equivalent fractions: $\frac{8}{56} + \frac{7}{56} = \frac{8 + 7}{56} = \frac{15}{56}$.
Step 4: The fraction $\frac{15}{56}$ is already in its simplest form.
Therefore, the solution to the problem is $\frac{15}{56}$.

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

• Step 1: Find the least common multiple (LCM) of the denominators 5 and 2. The LCM of 5 and 2 is 10.
• Step 2: Convert each fraction to have a denominator of 10.
• Step 3: Subtract the converted fractions.
• Step 4: Simplify the result if necessary.

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

Step 1: The LCM of 5 and 2 is 10, since 10 is the smallest number that both 5 and 2 divide into evenly.

Step 2: Convert each fraction to have a denominator of 10.

For $\frac{3}{5}$:
Multiply numerator and denominator by 2 to get $\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$.

For $\frac{1}{2}$:
Multiply numerator and denominator by 5 to get $\frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

Step 3: Subtract the fractions:

$\frac{6}{10} - \frac{5}{10} = \frac{6 - 5}{10} = \frac{1}{10}$.

Step 4: There is no further simplification needed for $\frac{1}{10}$ as it is already in its simplest form.

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

The correct answer, choice (4), is $\frac{1}{10}$.

To solve the problem of adding $\frac{2}{5}$ and $\frac{1}{6}$, we need to find a common denominator. We do this by multiplying the denominators: $5 \times 6 = 30$. This is the smallest common multiple of the two denominators and ensures that each fraction can be represented with a common base, allowing addition.

Let's convert each fraction to an equivalent fraction with the common denominator of 30:

• Convert $\frac{2}{5}$: Multiply both the numerator and the denominator by 6 to get $\frac{2 \times 6}{5 \times 6} = \frac{12}{30}$.

• Convert $\frac{1}{6}$: Multiply both the numerator and the denominator by 5 to get $\frac{1 \times 5}{6 \times 5} = \frac{5}{30}$.

Now, we add these equivalent fractions:

$\frac{12}{30} + \frac{5}{30} = \frac{12 + 5}{30} = \frac{17}{30}$.

The resulting fraction, $\frac{17}{30}$, is already in its simplest form because 17 is a prime number and does not share any common factors with 30 other than 1.

Thus, the sum of $\frac{2}{5}$ and $\frac{1}{6}$ is $\frac{17}{30}$.

Upon reviewing the given choices, the correct and matching choice is:

Choice 2: $\frac{17}{30}$

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

• Step 1: Find the common denominator for the fractions $\frac{2}{4}$ and $\frac{1}{3}$.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Perform the subtraction and simplify if necessary.

Now, let's work through these steps:

Step 1: The denominators are $4$ and $3$. The common denominator is the product $4 \times 3 = 12$.

Step 2: Convert each fraction:
$\frac{2}{4} = \frac{2 \times 3}{4 \times 3} = \frac{6}{12}$
$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$

Step 3: Subtract the fractions with a common denominator:
$\frac{6}{12} - \frac{4}{12} = \frac{6 - 4}{12} = \frac{2}{12}$

Finally, simplify $\frac{2}{12}$. The greatest common divisor of 2 and 12 is 2, so:
$\frac{2}{12} = \frac{2 \div 2}{12 \div 2} = \frac{1}{6}$

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

To solve the addition of fractions $\frac{3}{5} + \frac{1}{4}$, follow these steps:

• Step 1: Find a common denominator. The denominators are 5 and 4. The least common denominator is 20, which is the product of 5 and 4.
• Step 2: Convert each fraction to have the common denominator of 20.
  • For $\frac{3}{5}$, multiply both the numerator and the denominator by 4: $\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$.
  • For $\frac{1}{4}$, multiply both the numerator and denominator by 5: $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$.
• Step 3: Add the equivalent fractions: $\frac{12}{20} + \frac{5}{20} = \frac{12 + 5}{20} = \frac{17}{20}$.

Thus, the sum of $\frac{3}{5}$ and $\frac{1}{4}$ is $\frac{17}{20}$.

To solve the subtraction of fractions $\frac{3}{5} - \frac{1}{3}$, follow these steps:

• Step 1: Find the Least Common Denominator (LCD)
  The denominators are 5 and 3. The least common multiple of 5 and 3 is 15. Thus, the common denominator will be 15.
• Step 2: Convert fractions to have the same denominator
  For $\frac{3}{5}$, multiply both the numerator and the denominator by 3 to get:
  $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$.
  For $\frac{1}{3}$, multiply both the numerator and the denominator by 5 to get:
  $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Subtract the numerators
  Now subtract the equivalent fractions:
  $\frac{9}{15} - \frac{5}{15} = \frac{9 - 5}{15} = \frac{4}{15}$.
• Step 4: Simplify the fraction
  The fraction $\frac{4}{15}$ is already in its simplest form.

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

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

• Step 1: Identify the fractions and denominators involved.
• Step 2: Find a common denominator for the two fractions.
• Step 3: Convert each fraction to an equivalent fraction with the common denominator.
• Step 4: Subtract the numerators and simplify the result.

Let's perform each of these steps:

Step 1: We have the fractions $\frac{3}{7}$ and $\frac{1}{3}$.

Step 2: Find a common denominator. The denominators are 7 and 3, so the common denominator will be $7 \times 3 = 21$.

Step 3: Convert each fraction:

• For $\frac{3}{7}$, multiply both numerator and denominator by 3 to get $\frac{3 \times 3}{7 \times 3} = \frac{9}{21}$.
• For $\frac{1}{3}$, multiply both numerator and denominator by 7 to get $\frac{1 \times 7}{3 \times 7} = \frac{7}{21}$.

Step 4: Subtract the numerators:

$\frac{9}{21} - \frac{7}{21} = \frac{9 - 7}{21} = \frac{2}{21}$.

Simplify if necessary: Here, $\frac{2}{21}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{2}{21}$.

To solve $\frac{6}{10} - \frac{1}{3}$, we need to perform the following steps:

• Step 1: Identify a common denominator. The denominators 10 and 3 have a least common multiple of 30. We will use 30 as the common denominator.
• Step 2: Convert the fractions to have the common denominator of 30.
  $\frac{6}{10}$ needs to be converted by multiplying both the numerator and the denominator by 3: $\frac{6 \times 3}{10 \times 3} = \frac{18}{30}$ $\frac{1}{3}$ needs to be converted by multiplying both the numerator and the denominator by 10: $\frac{1 \times 10}{3 \times 10} = \frac{10}{30}$
• Step 3: Subtract the new fractions: $\frac{18}{30} - \frac{10}{30} = \frac{18 - 10}{30} = \frac{8}{30}$
• Step 4: Simplify the fraction $\frac{8}{30}$.
  Divide both the numerator and the denominator by their greatest common divisor, which is 2: $\frac{8 \div 2}{30 \div 2} = \frac{4}{15}$

Therefore, the result of the subtraction is $\frac{4}{15}$.

To solve the problem of adding the fractions $\frac{1}{2}$ and $\frac{2}{5}$, we will follow these steps:

• Step 1: Determine a common denominator for the fractions.
• Step 2: Convert each fraction to an equivalent fraction with this common denominator.
• Step 3: Add the resulting fractions.

Now, let’s explore each step in detail:

Step 1: The denominators are 2 and 5. A common denominator can be found by multiplying these two numbers: $2 \times 5 = 10$. Therefore, 10 is our common denominator.

Step 2: Convert each fraction to have the common denominator of 10.
- For $\frac{1}{2}$, multiply both the numerator and the denominator by 5:
$\frac{1}{2} \times \frac{5}{5} = \frac{5}{10}$.
- For $\frac{2}{5}$, multiply both the numerator and the denominator by 2:
$\frac{2}{5} \times \frac{2}{2} = \frac{4}{10}$.

Step 3: Add the fractions $\frac{5}{10}$ and $\frac{4}{10}$:
Combine the numerators while keeping the common denominator:
$5 + 4 = 9$.
Thus, $\frac{5}{10} + \frac{4}{10} = \frac{9}{10}$.

Therefore, the sum of $\frac{1}{2}$ and $\frac{2}{5}$ is $\frac{9}{10}$.

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

• Step 1: Simplify the fractions if possible.
• Step 2: Identify the common denominator.
• Step 3: Convert each fraction to have this common denominator.
• Step 4: Add the fractions.
• Step 5: Simplify the result, if necessary.

Step 1: Simplify $\frac{2}{4}$. It simplifies to $\frac{1}{2}$.

Step 2: The denominators are now 3 and 2. Find the least common multiple of 3 and 2, which is 6.

Step 3: Convert each fraction to have the common denominator of 6:
$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

Step 4: Add the fractions:
$\frac{2}{6} + \frac{3}{6} = \frac{2 + 3}{6} = \frac{5}{6}$

Step 5: The fraction $\frac{5}{6}$ is already in its simplest form.

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