Addition of Fractions: One of the denominators is the common denominator

Let's begin by identifying the lowest common denominator between 3 and 6.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 3 and 6.

In this case, the common denominator is 6.

Let's proceed to multiply each fraction by the appropriate number in order to reach the denominator 6.

We'll multiply the first fraction by 2

We'll multiply the second fraction by 1

$\frac{2\times2}{3\times2}+\frac{1\times1}{6\times1}=\frac{4}{6}+\frac{1}{6}$

Finally we'll combine and obtain the following result:

$\frac{4+1}{6}=\frac{5}{6}$

Let's first identify the lowest common denominator between 2 and 8.

In order to determine the lowest common denominator, we need to first find a number that is divisible by both 2 and 8.

In this case, the common denominator is 8.

We'll then proceed to multiply each fraction by the appropriate number in order to reach the denominator 8.

We'll multiply the first fraction by 4

We'll multiply the second fraction by 1

$\frac{1\times4}{2\times4}+\frac{3\times1}{8\times1}=\frac{4}{8}+\frac{3}{8}$

Finally we'll combine and obtain the following:

$\frac{4+3}{8}=\frac{7}{8}$

Let's first identify the lowest common denominator between 4 and 2.

In order to identify the lowest common denominator, we need to find a number that is divisible by both 4 and 2.

In this case, the common denominator is 4

We will then proceed to multiply each fraction by the appropriate number in order to reach the denominator 4

We'll multiply the first fraction by 1

We'll multiply the second fraction by 2

$\frac{2\times1}{4\times1}+\frac{1\times2}{2\times2}=\frac{2}{4}+\frac{2}{4}$

Finally we will combine and obtain the following:

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

Let's try to find the least common denominator between 2 and 10

To find the least common denominator, we need to find a number that is divisible by both 2 and 10

In this case, the common denominator is 10

Now we'll multiply each fraction by the appropriate number to reach the denominator 10

We'll multiply the first fraction by 5

We'll multiply the second fraction by 1

$\frac{1\times5}{2\times5}+\frac{3\times1}{10\times1}=\frac{5}{10}+\frac{3}{10}$

Now we'll combine and get:

$\frac{5+3}{10}=\frac{8}{10}$

Let's try to find the least common denominator between 4 and 8

To find the least common denominator, we need to find a number that is divisible by both 4 and 8

In this case, the common denominator is 8

Now we'll multiply each fraction by the appropriate number to reach the denominator 8

We'll multiply the first fraction by 2

We'll multiply the second fraction by 1

$\frac{1\times2}{4\times2}+\frac{6\times1}{8\times1}=\frac{2}{8}+\frac{6}{8}$

Now we'll combine and get:

$\frac{2+6}{8}=\frac{8}{8}=1$

Let's try to find the lowest common denominator between 2 and 10

To find the lowest common denominator, we need to find a number that is divisible by both 2 and 10

In this case, the common denominator is 10

Now we'll multiply each fraction by the appropriate number to reach the denominator 10

We'll multiply the first fraction by 5

We'll multiply the second fraction by 1

$\frac{1\times5}{2\times5}+\frac{2\times1}{10\times1}=\frac{5}{10}+\frac{2}{10}$

Now we'll combine and get:

$\frac{5+2}{10}=\frac{7}{10}$

Let's try to find the lowest common denominator between 3 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 3 and 9

In this case, the common denominator is 9

Now we'll multiply each fraction by the appropriate number to reach the denominator 9

We'll multiply the first fraction by 3

We'll multiply the second fraction by 1

$\frac{1\times3}{3\times3}+\frac{5\times1}{9\times1}=\frac{3}{9}+\frac{5}{9}$

Now we'll combine and get:

$\frac{3+5}{9}=\frac{8}{9}$

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

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

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 3

We'll multiply the second fraction by 1

$\frac{1\times3}{5\times3}+\frac{2\times1}{15\times1}=\frac{3}{15}+\frac{2}{15}$

Now we'll combine and get:

$\frac{3+2}{15}=\frac{5}{15}$

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

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

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 3

We'll multiply the second fraction by 1

$\frac{3\times3}{5\times3}+\frac{2\times1}{15\times1}=\frac{9}{15}+\frac{2}{15}$

Now we'll combine and get:

$\frac{9+2}{15}=\frac{11}{15}$

To solve $\frac{3}{4} + \frac{2}{20}$, let's follow these steps:

• Step 1: Convert $\frac{3}{4}$ to a fraction with the denominator $20$.
  Multiply both the numerator and denominator of $\frac{3}{4}$ by $5$, as $20 \div 4 = 5$:
  We have: $\frac{3 \times 5}{4 \times 5} = \frac{15}{20}$
• Step 2: The fraction $\frac{2}{20}$ is already in the correct form with a denominator of $20$.
• Step 3: Add the fractions $\frac{15}{20}$ and $\frac{2}{20}$:
  Since they have the same denominator, we combine the numerators directly: $\frac{15}{20} + \frac{2}{20} = \frac{15 + 2}{20} = \frac{17}{20}$

Therefore, the sum of the fractions is $\frac{17}{20}$.

To solve the problem, follow these steps:

• Step 1: Recognize the denominators $2$ and $10$ and note that $10$ is a common multiple.
• Step 2: Convert $\frac{1}{2}$ to a fraction with a denominator of 10. Since $2 \times 5 = 10$, multiply both the numerator and denominator of $\frac{1}{2}$ by 5 to get $\frac{5}{10}$.
• Step 3: The second fraction, $\frac{3}{10}$, already has a denominator of 10, so no conversion is needed.
• Step 4: Add the two fractions: $\frac{5}{10} + \frac{3}{10}$.
• Step 5: As the denominators are the same, add the numerators: $5 + 3 = 8$.
• Step 6: Write the result with the common denominator: $\frac{8}{10}$.
• Step 7: Check if the fraction can be simplified further. Since 8 and 10 have a common factor of 2, divide by 2 to simplify: $\frac{8}{10} = \frac{4}{5}$.

The problem's correct answer without simplification matches choice 1.

Therefore, the solution to the problem is $\frac{8}{10}$.

To solve the problem of adding $\frac{1}{6} + \frac{4}{12}$, follow these steps:

• Step 1: Find the least common denominator (LCD) for the fractions. The denominators are 6 and 12, and the LCD is 12.
• Step 2: Convert $\frac{1}{6}$ to an equivalent fraction with a denominator of 12. To do this, multiply both the numerator and the denominator by 2: $\frac{1}{6} \times \frac{2}{2} = \frac{2}{12}$.
• Step 3: The fraction $\frac{4}{12}$ already has the denominator of 12, so no conversion is needed.
• Step 4: Add the fractions with the common denominator: $\frac{2}{12} + \frac{4}{12} = \frac{6}{12}$.
• Step 5: Simplify the resulting fraction if possible. In this case, $\frac{6}{12}$ simplifies to $\frac{1}{2}$, but we will leave it as $\frac{6}{12}$ as it appears in the options.

Therefore, the solution to the problem is $\frac{6}{12}$, which matches option 3 from the provided answers.

To solve the problem of adding $\frac{2}{3}$ and $\frac{1}{6}$, we will first find a common denominator:

• Step 1: Identify the least common denominator (LCD), which is 6 for the fractions $\frac{2}{3}$ and $\frac{1}{6}$.
• Step 2: Convert $\frac{2}{3}$ to an equivalent fraction with a denominator of 6. To do this, multiply both the numerator and the denominator by 2: $\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$
• Step 3: Now, add the fractions $\frac{4}{6}$ and $\frac{1}{6}$: $\frac{4}{6} + \frac{1}{6} = \frac{4 + 1}{6} = \frac{5}{6}$

As we see, both fractions have been added correctly. The sum is already in its simplest form.

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

To solve the problem of adding $\frac{2}{3}$ and $\frac{1}{9}$, we follow these steps:

• Step 1: Find a common denominator.
  The denominators are 3 and 9. Since 9 is a multiple of 3, we can use 9 as the common denominator.
• Step 2: Convert the fractions to have the common denominator.
  - To convert $\frac{2}{3}$ to have a denominator of 9, multiply both the numerator and denominator by 3: $\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}$
• Step 3: Add the fractions $\frac{6}{9}$ and $\frac{1}{9}$.
  $\frac{6}{9} + \frac{1}{9} = \frac{6+1}{9} = \frac{7}{9}$
• Step 4: Simplify the result, if necessary.
  The fraction $\frac{7}{9}$ is already in its simplest form.

Therefore, the solution to $\frac{2}{3} + \frac{1}{9}$ is $\frac{7}{9}$.

To solve the problem of adding $\frac{3}{5} + \frac{4}{15}$, we follow these steps:

• Step 1: Identify a common denominator for the fractions. Since 15 is a multiple of 5, we can use 15 as the common denominator.
• Step 2: Convert $\frac{3}{5}$ to an equivalent fraction with a denominator of 15.
• Step 3: Add the fractions once they have a common denominator.
• Step 4: Simplify the result if possible.

Let's solve each step:
Step 1: Our common denominator is 15.
Step 2: To convert $\frac{3}{5}$ to a fraction with a denominator of 15, multiply both the numerator and the denominator by 3:
$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$.
Step 3: Now add $\frac{9}{15}$ and $\frac{4}{15}$:
$\frac{9}{15} + \frac{4}{15} = \frac{9 + 4}{15} = \frac{13}{15}$.
Step 4: The fraction $\frac{13}{15}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{13}{15}$.

To solve the problem of adding $\frac{3}{8}$ and $\frac{1}{4}$, we'll follow these steps:

• Step 1: Determine the Least Common Denominator (LCD).
• Step 2: Convert fractions to the common denominator.
• Step 3: Add the fractions.

Let's work through each step:
Step 1: The denominators are 8 and 4. The LCD of 8 and 4 is 8, as 8 is the smallest number that both 8 and 4 divide into without a remainder.

Step 2: Convert each fraction to have the common denominator 8.
- The fraction $\frac{3}{8}$ already has the denominator 8.
- Convert $\frac{1}{4}$ to a fraction with denominator 8: $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.

Step 3: Add the fractions $\frac{3}{8}$ and $\frac{2}{8}$:
The sum is $\frac{3 + 2}{8} = \frac{5}{8}$.

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

To solve the problem of adding $\frac{4}{15} + \frac{2}{5}$, follow these steps:

• Step 1: Identify a common denominator. Since 5 is a factor of 15, we will use 15 as the common denominator.
• Step 2: Convert $\frac{2}{5}$ to a fraction with a denominator of 15. To do this, multiply both the numerator and denominator by 3: $\frac{2 \times 3}{5 \times 3} = \frac{6}{15}$.
• Step 3: Add the fractions: $\frac{4}{15} + \frac{6}{15}$.
• Step 4: Since both fractions now have the same denominator, add the numerators: $\frac{4 + 6}{15} = \frac{10}{15}$.
• Step 5: Simplify $\frac{10}{15}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5: $\frac{10 \div 5}{15 \div 5} = \frac{2}{3}$.

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

To solve the problem of adding the fractions $\frac{1}{4}$ and $\frac{1}{8}$, follow these steps:

• Step 1: Identify the denominators: 4 and 8. We need a common denominator to add the fractions.
• Step 2: Find the least common multiple (LCM) of 4 and 8. The smallest number that both 4 and 8 can divide is 8, so our common denominator will be 8.
• Step 3: Convert $\frac{1}{4}$ to a fraction with denominator 8. To do this, multiply both the numerator and the denominator of $\frac{1}{4}$ by 2:
• $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
• Step 4: Now add the fractions: $\frac{2}{8} + \frac{1}{8}$.
• Since the fractions now have the same denominator, add the numerators: $2 + 1 = 3$, while keeping the denominator 8.
• The result is $\frac{3}{8}$.
• Step 5: Ensure the fraction is in its simplest form. The fraction $\frac{3}{8}$ is already simplified, as 3 and 8 have no common factors other than 1.

Therefore, the sum of $\frac{1}{4} + \frac{1}{8}$ is $\frac{3}{8}$.

Once we compare this with the given answer choices, we find that our final result, $\frac{3}{8}$, matches choice 1.

Hence, the correct answer to the problem is $\frac{3}{8}$.

To solve the problem of adding the fractions $\frac{7}{12}$ and $\frac{3}{4}$, we will follow these steps:

• Step 1: Find a common denominator.

• Step 2: Convert $\frac{3}{4}$ to an equivalent fraction with the denominator $12$.

• Step 3: Add the numerators of the fractions.

• Step 4: Simplify the resultant fraction if possible.

Now, let's perform the calculations:

Step 1: The denominator $12$ is already a common denominator for $\frac{7}{12}$, but we need to convert $\frac{3}{4}$ to have the same denominator. Since $4 \times 3 = 12$, multiply both the numerator and the denominator of $\frac{3}{4}$ by $3$:

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Step 2: Now, add $\frac{7}{12}$ and $\frac{9}{12}$:

$\frac{7}{12} + \frac{9}{12} = \frac{7+9}{12} = \frac{16}{12}$

Step 3: Simplify $\frac{16}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $4$:

$\frac{16}{12} = \frac{16 \div 4}{12 \div 4} = \frac{4}{3}$

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

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

• Step 1: Convert $\frac{1}{2}$ to have a denominator of 4. Multiply the numerator and the denominator by 2 to get $\frac{2}{4}$.
• Step 2: With both fractions now having a common denominator, add them: $\frac{2}{4} + \frac{2}{4}$.
• Step 3: Combine the numerators and place the sum over the common denominator: $\frac{2+2}{4} = \frac{4}{4}$.
• Step 4: Simplify the fraction $\frac{4}{4}$ to $1$.

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