Addition of Fractions: Finding a Common Denominator by Multiplying the Denominators

To solve this problem, we will add the fractions $\frac{1}{3}$ and $\frac{1}{10}$ by finding a common denominator.

• Step 1: Find a common denominator.
  Since the denominators are 3 and 10, the least common multiple (LCM) of these numbers is 30. Therefore, the common denominator will be 30.
• Step 2: Convert each fraction to have the common denominator.
  Convert $\frac{1}{3}$ into an equivalent fraction with a denominator of 30:
  $\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}$.
  Convert $\frac{1}{10}$ into an equivalent fraction with a denominator of 30:
  $\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}$.
• Step 3: Add the equivalent fractions.
  Now that both fractions have the same denominator, add the numerators while keeping the denominator 30:
  $\frac{10}{30} + \frac{3}{30} = \frac{10 + 3}{30} = \frac{13}{30}$.

After calculating, we find that the sum of the fractions is $\frac{13}{30}$.

Therefore, the correct answer to the problem is $\frac{13}{30}$.

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 of adding two fractions $\frac{1}{2}$ and $\frac{2}{7}$, follow these steps:

• Step 1: Determine the common denominator.

The denominators of the fractions are $2$ and $7$. Multiply these two numbers to find the common denominator: $2 \times 7 = 14$.

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

Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of $14$:
$\frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}$

Convert $\frac{2}{7}$ to an equivalent fraction with a denominator of $14$:
$\frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14}$

• Step 3: Add the adjusted fractions.

Now that both fractions have a common denominator, add them:
$\frac{7}{14} + \frac{4}{14} = \frac{7 + 4}{14} = \frac{11}{14}$

We have successfully added the fractions and obtained the result.

Therefore, the solution to the problem is $\frac{11}{14}$.

Let's try to find the lowest common denominator between 5 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 3

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 5

$\frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}$

Now we'll combine and get:

$\frac{6+5}{15}=\frac{11}{15}$

Let's try to find the lowest common denominator between 2 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 2 and 9

In this case, the common denominator is 18

Now we'll multiply each fraction by the appropriate number to reach the denominator 18

We'll multiply the first fraction by 9

We'll multiply the second fraction by 2

$\frac{1\times9}{2\times9}+\frac{2\times2}{9\times2}=\frac{9}{18}+\frac{4}{18}$

Now we'll combine and get:

$\frac{9+4}{18}=\frac{13}{18}$

To solve the addition of the fractions $\frac{2}{9}$ and $\frac{1}{2}$, follow these steps:

• Step 1: Determine the Common Denominator.
  The least common denominator for 9 and 2 is $18$ because $9 \times 2 = 18$.
• Step 2: Adjust Each Fraction.
  Convert $\frac{2}{9}$ to a fraction over 18. Multiply both the numerator and the denominator by 2:
  $\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}$.
  Convert $\frac{1}{2}$ to a fraction over 18. Multiply both the numerator and the denominator by 9:
  $\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}$.
• Step 3: Add the Fractions.
  Add the resulting fractions: $\frac{4}{18} + \frac{9}{18} = \frac{4 + 9}{18} = \frac{13}{18}$.

Thus, the sum of $\frac{2}{9}$ and $\frac{1}{2}$ is $\frac{13}{18}$.

To solve this problem, we will add the fractions $\frac{1}{2}$ and $\frac{1}{9}$ by finding a common denominator.

• First, identify the denominators: 2 and 9.
• Find a common denominator by multiplying the denominators: $2 \times 9 = 18$.
• Convert each fraction to an equivalent fraction with this common denominator:
• Convert $\frac{1}{2}$ to have a denominator of 18 by multiplying both the numerator and denominator by 9: $\frac{1 \times 9}{2 \times 9} = \frac{9}{18}$.
• Convert $\frac{1}{9}$ to have a denominator of 18 by multiplying both the numerator and denominator by 2: $\frac{1 \times 2}{9 \times 2} = \frac{2}{18}$.
• Add the converted fractions: $\frac{9}{18} + \frac{2}{18} = \frac{11}{18}$.
• The fraction $\frac{11}{18}$ is already in its simplest form.

Thus, the sum of the fractions $\frac{1}{2}$ and $\frac{1}{9}$ is $\frac{11}{18}$.

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 $\frac{13}{20}$.

Let's try to find the least common denominator between 5 and 3

To find the least common denominator, we need to find a number that is divisible by both 5 and 3

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 5

$\frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}$

Now we'll combine and get:

$\frac{6+5}{15}=\frac{11}{15}$

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 $\frac{13}{15}$.

Let's try to find the lowest common denominator between 7 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 7 and 3

In this case, the common denominator is 21

Now we'll multiply each fraction by the appropriate number to reach the denominator 21

We'll multiply the first fraction by 3

We'll multiply the second fraction by 7

$\frac{1\times3}{7\times3}+\frac{1\times7}{3\times7}=\frac{3}{21}+\frac{7}{21}$

Now we'll combine and get:

$\frac{3+7}{21}=\frac{10}{21}$

Let's try to find the lowest common denominator between 4 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 9

In this case, the common denominator is 36

Now we'll multiply each fraction by the appropriate number to reach the denominator 36

We'll multiply the first fraction by 9

We'll multiply the second fraction by 4

$\frac{1\times9}{4\times9}+\frac{1\times4}{9\times4}=\frac{9}{36}+\frac{4}{36}$

Now we'll combine and get:

$\frac{9+4}{36}=\frac{13}{36}$

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 $\frac{3}{2}$.

Let's try to find the lowest common denominator between 5 and 4

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 4

In this case, the common denominator is 20

Now we'll multiply each fraction by the appropriate number to reach the denominator 20

We'll multiply the first fraction by 4

We'll multiply the second fraction by 5

$\frac{2\times4}{5\times4}+\frac{1\times5}{4\times5}=\frac{8}{20}+\frac{5}{20}$

Now we'll combine and get:

$\frac{8+5}{20}=\frac{13}{20}$

To solve the problem of adding $\frac{2}{8}$ and $\frac{1}{3}$, we need to first convert these fractions to have a common denominator.

Step 1: Find the least common denominator (LCD).
- The denominators of the fractions are $8$ and $3$.
- The common denominator can be found by multiplying $8$ and $3$, which gives us $24$.

Step 2: Convert each fraction to an equivalent fraction with the common denominator of $24$.
- For $\frac{2}{8}$, multiply both the numerator and the denominator by $3$:
$\frac{2}{8} = \frac{2 \times 3}{8 \times 3} = \frac{6}{24}$.
- For $\frac{1}{3}$, multiply both the numerator and the denominator by $8$:
$\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$.

Step 3: Add the resulting fractions.
- $\frac{6}{24} + \frac{8}{24} = \frac{6 + 8}{24} = \frac{14}{24}$.

Therefore, the solution to the problem is $\frac{14}{24}$, which simplifies our answer.

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 first find a common denominator for $\frac{2}{11}$ and $\frac{1}{2}$. The denominators are 11 and 2, and their product gives a common denominator of $22$.

Next, we adjust each fraction:

• $\frac{2}{11}$ is adjusted to $\frac{2 \times 2}{22} = \frac{4}{22}$.
• $\frac{1}{2}$ is adjusted to $\frac{1 \times 11}{22} = \frac{11}{22}$.

Now, add the adjusted fractions:

$\frac{4}{22} + \frac{11}{22} = \frac{4 + 11}{22} = \frac{15}{22}$

Therefore, the solution to the problem is $\frac{15}{22}$.

The correct answer from the choices provided is $\frac{15}{22}$.