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

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

• Step 1: Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of 12.
  Since $4 \times 3 = 12$, multiply both the numerator and the denominator of $\frac{1}{4}$ by 3 to get $\frac{3}{12}$.
• Step 2: Add $\frac{3}{12} + \frac{3}{12}$.
  The numerators will add to $3 + 3$, giving $\frac{6}{12}$.
• Step 3: Simplify the fraction $\frac{6}{12}$.
  The greatest common divisor of 6 and 12 is 6, so divide both the numerator and the denominator by 6 to get $\frac{1}{2}$.

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

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 $1$.

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 $\frac{13}{15}$.

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 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 $\frac{3}{4} + \frac{2}{20}$, 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 this problem, we need to add the fractions $\frac{3}{5}$ and $\frac{6}{10}$. Since $\frac{6}{10}$ is already expressed with the denominator of 10, we will convert $\frac{3}{5}$ to have the same denominator.

Step 1: Convert $\frac{3}{5}$ into a fraction with a denominator of 10. To do this, multiply both the numerator and the denominator by 2:

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

Step 2: Add the fractions $\frac{6}{10}$ and $\frac{6}{10}$:

$\frac{6}{10} + \frac{6}{10} = \frac{6 + 6}{10} = \frac{12}{10}$

Step 3: Simplify $\frac{12}{10}$. Both numerator and denominator can be divided by 2:

$\frac{12}{10} = \frac{12 \div 2}{10 \div 2} = \frac{6}{5}$

Thus, the sum $\frac{3}{5} + \frac{6}{10}$ simplifies to $\frac{6}{5}$.

Therefore, the correct answer is $\frac{6}{5}$ which corresponds to choice 3.

To solve this problem, we need to add the fractions $\frac{1}{2}$ and $\frac{3}{8}$.

• Step 1: Convert $\frac{1}{2}$ to a fraction with a denominator of 8. We do this by determining the equivalent fraction $\frac{1}{2} = \frac{4}{8}$. We achieve this by multiplying the numerator and denominator by 4.
• Step 2: Now, add the fractions $\frac{4}{8}$ and $\frac{3}{8}$: $\frac{4}{8} + \frac{3}{8} = \frac{4 + 3}{8} = \frac{7}{8}$ This step involves adding the numerators while keeping the common denominator.

Therefore, the sum of $\frac{1}{2}$ and $\frac{3}{8}$ is $\frac{7}{8}$.

To solve this problem, we need to add the fractions $\frac{1}{18}$ and $\frac{1}{6}$:

Step 1: Find the least common denominator
The denominators are 18 and 6. The least common multiple of 18 and 6 is 18.

Step 2: Convert each fraction to have the least common denominator
The fraction $\frac{1}{18}$ already has the denominator 18, so it remains $\frac{1}{18}$.
To convert $\frac{1}{6}$ to a fraction with denominator 18, multiply both the numerator and denominator by 3: $\frac{1 \times 3}{6 \times 3} = \frac{3}{18}$.

Step 3: Add the converted fractions
Now that both fractions have the same denominator, add them:
$\frac{1}{18} + \frac{3}{18} = \frac{1 + 3}{18} = \frac{4}{18}$.

Step 4: Simplify the resulting fraction
$\frac{4}{18}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
$\frac{4 \div 2}{18 \div 2} = \frac{2}{9}$.

Therefore, the sum of $\frac{1}{18}$ and $\frac{1}{6}$ is $\frac{2}{9}$.

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 $\frac{4}{15} + \frac{2}{5}$, 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 this problem, we'll follow these steps:

• Step 1: Identify the least common denominator (LCD) for the given fractions.
• Step 2: Convert each fraction to an equivalent fraction with this common denominator.
• Step 3: Add the numerators, keeping the denominator unchanged.
• Step 4: Simplify the resultant fraction if possible.

Now, let's work through each step:

Step 1: Identify the least common denominator (LCD).
The denominators are 6 and 12. The smallest number that both 6 and 12 divide evenly into is 12. Therefore, the LCD is 12.

Step 2: Convert the fractions to have the LCD as their denominator.
$\frac{2}{6}$ needs to be converted to a fraction with a denominator of 12. We multiply both the numerator and denominator by 2:

$\frac{2 \times 2}{6 \times 2} = \frac{4}{12}$.

The second fraction, $\frac{5}{12}$, already has the denominator of 12, so it remains $\frac{5}{12}$.

Step 3: Add the two fractions:
$\frac{4}{12} + \frac{5}{12} = \frac{4 + 5}{12} = \frac{9}{12}$.

Step 4: Simplify the fraction.
The fraction $\frac{9}{12}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{9 \div 3}{12 \div 3} = \frac{3}{4}$.

Therefore, after fully simplifying, the sum of the fractions is $\frac{3}{4}$.

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 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:

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

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

• Step 1: Identify the least common denominator (LCD) of the fractions $\frac{1}{4}$ and $\frac{1}{2}$.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Add the fractions together.

Now, let's work through each step:

Step 1: Identify the Least Common Denominator (LCD).

The denominators are 4 and 2. The smallest number that both 4 and 2 can divide into without a remainder is 4. Thus, the LCD is 4.

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

The fraction $\frac{1}{4}$ already has the denominator 4, so it remains the same: $\frac{1}{4}$.

The fraction $\frac{1}{2}$ needs to be converted. We multiply both the numerator and denominator by 2 to get the equivalent fraction $\frac{2}{4}$.

Step 3: Add the fractions.

The fractions $\frac{1}{4}$ and $\frac{2}{4}$ share a common denominator, so we can add the numerators:

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

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

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:

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

We must first identify 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.

We'll then proceed to multiply each fraction by the appropriate number to reach the denominator 6.

We'll multiply the first fraction by 2

We'll multiply the second fraction by 1

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

Finally we'll combine and obtain the following:

$\frac{2+3}{6}=\frac{5}{6}$

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 this problem, we'll follow these steps:

• Identify the common denominator.
• Convert each fraction to an equivalent fraction with the common denominator.
• Add the fractions.
• Verify the solution against given choices.

Now, let's work through each step:

Step 1: Identify the common denominator. For fractions $\frac{1}{3}$ and $\frac{1}{6}$, the least common multiple (LCM) of 3 and 6 is 6.

Step 2: Convert $\frac{1}{3}$ to have a denominator of 6. We do this by multiplying both the numerator and denominator by 2:

$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

The fraction $\frac{1}{6}$ already has a denominator of 6, so we leave it unchanged:

$\frac{1}{6} = \frac{1}{6}$

Step 3: Add the fractions:

$\frac{2}{6} + \frac{1}{6} = \frac{2 + 1}{6} = \frac{3}{6}$

The fraction $\frac{3}{6}$ simplifies to $\frac{1}{2}$, but since the task is to match with given choices, we note that there is no need to simplify further.

After comparing with the given choices, the option that matches our calculation is:

$\frac{3}{6}$

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 $\frac{5}{8}$.