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

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 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 the problem of adding the fractions $\frac{1}{5}$ and $\frac{1}{3}$, we follow these steps:

• Step 1: Find a common denominator for the fractions. Since the denominators are $5$ and $3$, the least common multiple is $15$.
• Step 2: Convert each fraction to this common denominator:
  - For $\frac{1}{5}$, multiply both numerator and denominator by $3$ (the denominator of the other fraction), resulting in $\frac{3}{15}$.
  - For $\frac{1}{3}$, multiply both numerator and denominator by $5$ (the denominator of the other fraction), resulting in $\frac{5}{15}$.
• Step 3: Add the fractions now that they have a common denominator:
  $\frac{3}{15} + \frac{5}{15} = \frac{3+5}{15} = \frac{8}{15}$.

Therefore, when you add $\frac{1}{5}$ and $\frac{1}{3}$, the solution is $\frac{8}{15}$.

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 $\frac{13}{20}$.

Let's try to find the least common multiple (LCM) between 8 and 3

To find the least common multiple, we need to find a number that is divisible by both 8 and 3

In this case, the common multiple is 24

Now we'll multiply each fraction by the appropriate number to reach the denominator 24

We'll multiply the first fraction by 3

We'll multiply the second fraction by 8

$\frac{3\times3}{8\times3}+\frac{2\times8}{3\times8}=\frac{9}{24}+\frac{16}{24}$

Now we'll combine and get:

$\frac{9+16}{24}=\frac{25}{24}$

To solve the given problem, we will follow these steps:

• Step 1: Determine a common denominator for the fractions.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Add the numerators and keep the common denominator.
• Step 4: Simplify the resulting fraction if possible.

Let's proceed with each step:

Step 1: Determine a common denominator.
The denominators of the fractions are 5 and 7. The least common multiple (LCM) of 5 and 7 is 35. Thus, the common denominator is 35.

Step 2: Convert each fraction to have the common denominator of 35.
Convert $\frac{3}{5}$ to a fraction with a denominator of 35: $\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$.
Convert $\frac{2}{7}$ to a fraction with a denominator of 35: $\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}$.

Step 3: Add the numerators and use the common denominator.
Now add the fractions: $\frac{21}{35} + \frac{10}{35} = \frac{21+10}{35} = \frac{31}{35}$.

Step 4: Simplify the result.
The fraction $\frac{31}{35}$ is already in its simplest form since 31 and 35 have no common factors other than 1.

Therefore, the solution to the problem is $\frac{31}{35}$.

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

• Step 1: Identify a common denominator for the fractions. The current denominators are $5$ and $3$, hence their common denominator is $15$ (since $5 \times 3 = 15$).
• Step 2: Convert each fraction to an equivalent fraction with the common denominator $15$:
• For $\frac{4}{5}$: multiply the numerator and the denominator by $3$ (since $5 \times 3 = 15$).
  $\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}$.
• For $\frac{1}{3}$: multiply the numerator and the denominator by $5$ (since $3 \times 5 = 15$).
  $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Add the converted fractions:
  $\frac{12}{15} + \frac{5}{15} = \frac{12 + 5}{15} = \frac{17}{15}$.

Therefore, the solution to the problem is $\frac{17}{15}$.

To solve the problem of adding $\frac{4}{9}$ and $\frac{1}{2}$, we'll proceed step-by-step:

• Step 1: Determine a common denominator.
• Step 2: Convert each fraction to an equivalent fraction with this common denominator.
• Step 3: Add the numerators of these converted fractions.
• Step 4: Simplify the resulting fraction if necessary.

Now, let's perform these steps in detail:

Step 1: Determine the common denominator.
The denominators are 9 and 2. The least common denominator (LCD) can be found by multiplying these because they have no common factors other than 1:
$\text{LCD} = 9 \times 2 = 18$.

Step 2: Convert each fraction to have the common denominator of 18.

• Convert $\frac{4}{9}$: $\frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18}$
• Convert $\frac{1}{2}$: $\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}$

Step 3: Add the numerators of the converted fractions:
$\frac{8}{18} + \frac{9}{18} = \frac{8+9}{18} = \frac{17}{18}$

Step 4: Simplification (if needed):
The fraction $\frac{17}{18}$ is already in its simplest form.

Therefore, the sum of $\frac{4}{9}$ and $\frac{1}{2}$ is $\frac{17}{18}$.

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 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:

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

Let's try to find the lowest common denominator between 4 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 6

In this case, the common denominator is 12

Now we'll multiply each fraction by the appropriate number to reach the denominator 12

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{1\times3}{4\times3}+\frac{2\times2}{6\times2}=\frac{3}{12}+\frac{4}{12}$

Now we'll combine and get:

$\frac{3+4}{12}=\frac{7}{12}$

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 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:

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

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 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 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, 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 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 $\frac{5}{6}$.