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

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:

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

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 $\frac{5}{8}$.

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'll follow these steps:

• Step 1: Convert $\frac{1}{4}$ to a fraction with a denominator of 8.
• Step 2: Add the fractions.
• Step 3: Simplify the result.

Now, let's work through each step:

Step 1: Convert $\frac{1}{4}$ to have a denominator of 8. Since $8 \div 4 = 2$, multiply both the numerator and denominator of $\frac{1}{4}$ by 2:

$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$

Step 2: Now add $\frac{2}{8}$ and $\frac{4}{8}$:

$\frac{2}{8} + \frac{4}{8} = \frac{2 + 4}{8} = \frac{6}{8}$

Step 3: Simplify $\frac{6}{8}$ if possible. The greatest common divisor of 6 and 8 is 2. So, $\frac{6}{8}$ simplifies to:

$\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

Therefore, the solution to the problem is $\frac{6}{8}$.

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 this problem, we will follow these steps:

• Step 1: Identify the given fractions $\frac{1}{2}$ and $\frac{2}{8}$.
• Step 2: Determine a common denominator for both fractions. Since 8 is a multiple of 2, we will use 8 as the common denominator.
• Step 3: Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8.
• Step 4: Add the fractions together.

Now, let's work through these steps:

Step 1: The fractions given are $\frac{1}{2}$ and $\frac{2}{8}$.

Step 2: We choose 8 as the common denominator because it is a multiple of 2.

Step 3: Convert $\frac{1}{2}$ to have a denominator of 8. To do this, multiply both the numerator and the denominator of $\frac{1}{2}$ by 4:

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

The fractions now are $\frac{4}{8}$ and $\frac{2}{8}$, both having the common denominator 8.

Step 4: Add the numerators of these fractions:

$\frac{4}{8} + \frac{2}{8} = \frac{4+2}{8} = \frac{6}{8}$

Therefore, the sum of $\frac{1}{2} + \frac{2}{8}$ is $\frac{6}{8}$.

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 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 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 this problem, we'll follow these steps:

• Step 1: Convert the fractions to have the same denominator.
• Step 2: Add the fractions.
• Step 3: Simplify the resulting fraction.

Let's proceed with solving the problem:

Step 1: Convert $\frac{2}{5}$ to a fraction with a denominator of 10. The equivalent fraction is found by multiplying the numerator and the denominator by 2:

$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$

Step 2: Add the fractions $\frac{4}{10}$ and $\frac{5}{10}$:

$\frac{4}{10} + \frac{5}{10} = \frac{4 + 5}{10} = \frac{9}{10}$

Step 3: Simplify the resulting fraction. Since $\frac{9}{10}$ is already in its simplest form, we conclude:

Therefore, the solution to the problem is $\frac{9}{10}$.

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 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 this problem, we'll follow these steps:

• Step 1: Determine the least common denominator (LCD) of the fractions.

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

• Step 3: Add the numerators and keep the denominator the same.

Now, let's work through each step:

Step 1: The least common denominator of 15 and 5 is 15.

Step 2: Convert the fraction $\frac{2}{5}$ to have the denominator of 15.
To convert $\frac{2}{5}$, we need to multiply both the numerator and the denominator by 3, because ${5 \times 3 = 15}$.
Thus, $\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$.

Step 3: Now, add the fractions $\frac{1}{15}$ and $\frac{6}{15}$.
When adding these fractions, the equation is $\frac{1}{15} + \frac{6}{15} = \frac{1 + 6}{15} = \frac{7}{15}$.

Therefore, the solution to the problem is $\frac{7}{15}$.

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 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 $\frac{1}{2}$.

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 $\frac{4}{15} + \frac{2}{5}$, 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) of the fractions.
• Step 2: Rewrite the fractions with the common denominator.
• Step 3: Add the fractions.
• Step 4: Simplify the resulting fraction.

Let's work through these steps:

Step 1: The denominators of our fractions are 12 and 36. The least common denominator is 36. This is because 36 is the smallest number that both 12 and 36 divide into evenly.

Step 2: Rewrite $\frac{5}{12}$ with the denominator 36. To do this, find what number 12 must be multiplied by to become 36, which is 3. Thus, multiply both the numerator and the denominator of $\frac{5}{12}$ by 3:

$\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}$.

Step 3: Now add the fractions $\frac{15}{36}$ and $\frac{11}{36}$, since they have a common denominator:

$\frac{15}{36} + \frac{11}{36} = \frac{15 + 11}{36} = \frac{26}{36}$.

Step 4: Simplify $\frac{26}{36}$. The greatest common divisor (GCD) of 26 and 36 is 2. Divide both the numerator and the denominator by 2:

$\frac{26}{36} = \frac{26 \div 2}{36 \div 2} = \frac{13}{18}$.

Therefore, the sum of $\frac{5}{12} + \frac{11}{36}$ is $\frac{13}{18}$.

The correct choice that matches this solution is choice 4.

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:

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