Addition of Fractions: More than two fractions

To solve this problem, we'll add the fractions $\frac{2}{5}$, $\frac{1}{6}$, and $\frac{4}{30}$. Here's the step-by-step solution:

Step 1: Find a common denominator.

The denominators are 5, 6, and 30. The least common multiple (LCM) of these numbers is 30.

Step 2: Convert each fraction to have the common denominator of 30.

• $\frac{2}{5}$ is converted by multiplying both the numerator and the denominator by 6: $\frac{2 \times 6}{5 \times 6} = \frac{12}{30}$
• $\frac{1}{6}$ is converted by multiplying both the numerator and the denominator by 5: $\frac{1 \times 5}{6 \times 5} = \frac{5}{30}$
• $\frac{4}{30}$ already has the denominator of 30, so it remains $\frac{4}{30}$.

Step 3: Add the converted fractions.

Now, add the numerators while keeping the common denominator: $\frac{12}{30} + \frac{5}{30} + \frac{4}{30} = \frac{12 + 5 + 4}{30} = \frac{21}{30}$

Therefore, the sum of the fractions is $\frac{21}{30}$.

From the provided answer choices, the correct answer is choice 2: $\frac{21}{30}$.

To solve this problem, we will simplify and add the fractions $\frac{1}{2} + \frac{2}{4} + \frac{3}{6}$ step-by-step:

• Step 1: Simplify fractions where possible.
  $\frac{2}{4}$ simplifies to $\frac{1}{2}$ because the numerator and denominator can both be divided by 2.
  Similarly, $\frac{3}{6}$ simplifies to $\frac{1}{2}$ because both the numerator and denominator are divisible by 3.
• Step 2: Add the simplified fractions.
  The equation becomes $\frac{1}{2} + \frac{1}{2} + \frac{1}{2}$.
• Step 3: Since all fractions now have the same denominator, they can be added directly:
  Add the numerators: $1 + 1 + 1 = 3$ and keep the denominator the same: 2.
  This gives us $\frac{3}{2}$.

Therefore, the solution to the problem is $\frac{3}{2}$.

To solve the problem of adding the fractions $\frac{3}{10}+\frac{1}{5}+\frac{4}{6}$, we follow these steps:

• Find the common denominator: The denominators are 10, 5, and 6. The least common multiple (LCM) of these numbers is 30.
• Convert each fraction to have the denominator 30:
  - $\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}$
  - $\frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30}$
  - $\frac{4}{6} = \frac{4 \times 5}{6 \times 5} = \frac{20}{30}$
• Add the converted fractions together:
  - $\frac{9}{30} + \frac{6}{30} + \frac{20}{30} = \frac{9 + 6 + 20}{30} = \frac{35}{30}$
• Simplify the fraction $\frac{35}{30}$:
  - The greatest common divisor (GCD) of 35 and 30 is 5. Thus, $\frac{35}{30} = \frac{35 \div 5}{30 \div 5} = \frac{7}{6}$.

Therefore, the solution to the problem is $\frac{7}{6}$.

To solve this problem, we will add the fractions $\frac{1}{2}$, $\frac{3}{4}$, and $\frac{6}{8}$ by first converting each to have a common denominator:

Step 1: Find the Least Common Denominator (LCD). The denominators are $2$, $4$, and $8$. The LCM of these numbers is $8$.

Step 2: Rewrite each fraction with the denominator of $8$.

• $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$
• $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$
• $\frac{6}{8}$ is already expressed with a denominator of $8$.

Step 3: Add the fractions:

$\frac{4}{8} + \frac{6}{8} + \frac{6}{8} = \frac{4 + 6 + 6}{8} = \frac{16}{8}$

Step 4: Simplify the fraction:

$\frac{16}{8} = 2$

Therefore, the sum of the fractions $\frac{1}{2} + \frac{3}{4} + \frac{6}{8} = 2$.

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

• Step 1: Find the least common multiple (LCM) of the denominators: 5, 3, and 15.
• Step 2: Convert each fraction to an equivalent fraction with the LCM as the common denominator.
• Step 3: Add the numerators of the converted fractions.
• Step 4: Simplify the resulting fraction if necessary.

Now, let's work through each step:

Step 1: The denominators are 5, 3, and 15. The LCM of these numbers is 15.

Step 2: Convert each fraction:

• $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$
• $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$
• $\frac{2}{15} = \frac{2}{15}$

Step 3: Add the numerators of the converted fractions:

$\frac{9}{15} + \frac{5}{15} + \frac{2}{15} = \frac{9 + 5 + 2}{15} = \frac{16}{15}$

Step 4: The fraction $\frac{16}{15}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{16}{15}$.

To solve the problem, we start by finding the least common denominator (LCD) for the fractions $\frac{4}{12}$, $\frac{1}{3}$, and $\frac{1}{6}$.

The denominators are 12, 3, and 6. We need to find the smallest number that is a multiple of each of these numbers. The LCD of 12, 3, and 6 is 12.

Next, we convert each fraction to have this common denominator:

• $\frac{4}{12}$ already has 12 as the denominator.
• $\frac{1}{3}$ can be converted to have 12 as the denominator by multiplying both the numerator and the denominator by 4: $\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$.
• $\frac{1}{6}$ can be converted to have 12 as the denominator by multiplying both the numerator and the denominator by 2: $\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$.

Now, we simply add these fractions:

$\frac{4}{12} + \frac{4}{12} + \frac{2}{12} = \frac{4 + 4 + 2}{12} = \frac{10}{12}$.

Therefore, the solution to the problem is $\frac{10}{12}$.

To solve this problem, we need to add the fractions $\frac{1}{5}$, $\frac{3}{10}$, and $\frac{2}{5}$.

First, we need to find a common denominator for all the fractions. The denominators we have are 5, 10, and 5. The least common multiple (LCM) of these numbers is 10.

Let's convert each fraction to have a denominator of 10:

• $\frac{1}{5}$ can be converted to: $\frac{1 \times 2}{5 \times 2} = \frac{2}{10}$
• $\frac{3}{10}$ is already with a denominator of 10, so it remains: $\frac{3}{10}$
• $\frac{2}{5}$ can be converted to: $\frac{2 \times 2}{5 \times 2} = \frac{4}{10}$

Now we can add the fractions:

$\frac{2}{10} + \frac{3}{10} + \frac{4}{10} = \frac{2 + 3 + 4}{10} = \frac{9}{10}$

Therefore, the sum of the fractions is $\frac{9}{10}$.

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

We will add the fractions $\frac{1}{6}$, $\frac{1}{3}$, and $\frac{2}{12}$ by first finding the common denominator.

The least common denominator (LCD) of the denominators 6, 3, and 12 is 12.

Let's convert each fraction to have this common denominator:

• Convert $\frac{1}{6}$: Multiply both the numerator and the denominator by 2: $\frac{1 \cdot 2}{6 \cdot 2} = \frac{2}{12}$.
• Convert $\frac{1}{3}$: Multiply both the numerator and the denominator by 4: $\frac{1 \cdot 4}{3 \cdot 4} = \frac{4}{12}$.
• $\frac{2}{12}$ already has the denominator 12.

Now add the fractions:

$\frac{2}{12} + \frac{4}{12} + \frac{2}{12} = \frac{2 + 4 + 2}{12} = \frac{8}{12}$.

The fraction $\frac{8}{12}$ can be simplified, but since the problem specifies to provide the answer in this form, we leave it as is.

Therefore, the solution to the problem is $\frac{8}{12}$.

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

• Step 1: Find the least common denominator of 2, 8, and 4.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Perform the addition of the adjusted fractions.
• Step 4: Simplify the result if needed.

Now, let's work through each step:
Step 1: The denominators are 2, 8, and 4. The least common denominator (LCD) of these numbers is 8.
Step 2: Convert each fraction:
$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$
$\frac{1}{8} = \frac{1}{8}$ (already in the desired form)
$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
Step 3: Add the fractions:
$\frac{4}{8} + \frac{1}{8} + \frac{2}{8} = \frac{4 + 1 + 2}{8} = \frac{7}{8}$
Step 4: The fraction $\frac{7}{8}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{7}{8}$.

To solve this problem, let's proceed with the following steps:

• Step 1: Identify the common denominator for the fractions $\frac{1}{5}$, $\frac{3}{15}$, and $\frac{1}{3}$. The denominators are 5, 15, and 3. The LCM of these denominators is 15.
• Step 2: Convert each fraction to an equivalent fraction with 15 as the denominator.
  • $\frac{1}{5}$ becomes $\frac{1 \times 3}{5 \times 3} = \frac{3}{15}$.
  • $\frac{3}{15}$ already has a denominator of 15, so it remains $\frac{3}{15}$.
  • $\frac{1}{3}$ becomes $\frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Add the equivalent fractions: $\frac{3}{15} + \frac{3}{15} + \frac{5}{15}$.
  • Add the numerators: $3 + 3 + 5 = 11$.
  • The common denominator is 15, so the result is $\frac{11}{15}$.
• Step 4: Simplify the fraction if necessary. In this case, $\frac{11}{15}$ is already in its simplest form as 11 and 15 have no common factors other than 1.

Therefore, the solution to the problem is $\frac{11}{15}$.

To solve the given problem, let's follow these steps:

• Step 1: Find the Least Common Denominator (LCD)
  The denominators are 2, 6, and 12. The least common multiple (LCM) of these numbers is 12.
• Step 2: Convert each fraction to have the common denominator of 12
  Here is how to convert each fraction:
  • $\frac{1}{2}$ becomes $\frac{1 \times 6}{2 \times 6} = \frac{6}{12}$
  • $\frac{1}{6}$ becomes $\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$
  • $\frac{3}{12}$ is already with denominator 12, so it remains $\frac{3}{12}$.
• Step 3: Add the fractions
  Now add the equivalent fractions: $\frac{6}{12} + \frac{2}{12} + \frac{3}{12} = \frac{6 + 2 + 3}{12} = \frac{11}{12}$.

Therefore, the solution to the problem is $\frac{11}{12}$.

When we have a fraction addition exercise with more than one fraction, we make sure all the denominators of the fractions are identical.

Let's find the common denominator of the fractions' denominators: $10$ and $5$

The common denominator is $10$.

Now we'll multiply both numerator and denominator of the fraction $\frac{1}{5}$ by $2$ and create a fraction addition exercise where all denominators are $10$:

$\frac{3}{10}+\frac{1\times2}{5\times2}+\frac{4}{10}=\frac{3}{10}+\frac{2}{10}+\frac{4}{10}$

Finally, we'll add all the numerators of the fractions:

$\frac{3}{10}+\frac{2}{10}+\frac{4}{10}=\frac{3+2+4}{10}=\frac{9}{10}$

To solve this problem, we will follow these steps:

• Step 1: Find the least common multiple (LCM) of the denominators.
• Step 2: Convert each fraction to an equivalent fraction with the common denominator.
• Step 3: Add the converted fractions.
• Step 4: Simplify the resulting fraction if possible.

Step 1: The denominators are 6, 4, and 12. The least common multiple of these numbers is 12.

Step 2: Convert each fraction to have the common denominator of 12.
- Convert $\frac{2}{6}$ to have the denominator 12: $\frac{2}{6} = \frac{4}{12}$ by multiplying numerator and denominator by 2.
- Convert $\frac{1}{4}$ to have the denominator 12: $\frac{1}{4} = \frac{3}{12}$ by multiplying numerator and denominator by 3.
- $\frac{2}{12}$ already has the denominator 12, so it remains unchanged: $\frac{2}{12}$.

Step 3: Add the numerators of the converted fractions:
$\frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{4 + 3 + 2}{12} = \frac{9}{12}$.

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

Therefore, the solution to the problem is $\frac{9}{12}$.

First, let's find a common denominator for 4, 8, and 6: it's 24.

Now, we'll multiply each fraction by the appropriate number to get:

$\frac{5\times4}{6\times4}x+\frac{7\times3}{8\times3}x+\frac{2\times6}{4\times6}x=$

Let's solve the multiplication exercises in the numerator and denominator:

$\frac{20}{24}x+\frac{21}{24}x+\frac{12}{24}x=$

We'll connect all the numerators:

$\frac{20+21+12}{24}x=\frac{41+12}{24}x=\frac{53}{24}x$

Let's break down the numerator into a smaller addition exercise:

$\frac{48+5}{24}=\frac{48}{24}+\frac{5}{24}=2+\frac{5}{24}=2\frac{5}{24}x$

To solve the expression $\frac{1}{2} - \frac{2}{8} + \frac{1}{4}$, we must first find a common denominator for the fractions involved.

Step 1: Identify a common denominator. The denominators are 2, 8, and 4. The smallest common multiple of these numbers is 8.

Step 2: Convert each fraction to have the common denominator of 8.

• The fraction $\frac{1}{2}$ can be written as $\frac{4}{8}$ because $1 \times 4 = 4$ and $2 \times 4 = 8$.
• The fraction $\frac{2}{8}$ is already expressed with 8 as the denominator.
• The fraction $\frac{1}{4}$ can be written as $\frac{2}{8}$ because $1 \times 2 = 2$ and $4 \times 2 = 8$.

Step 3: Substitute these equivalent fractions back into the original expression:

$\frac{4}{8} - \frac{2}{8} + \frac{2}{8}$

Step 4: Perform the subtraction and addition following the order of operations:

• Subtract: $\frac{4}{8} - \frac{2}{8} = \frac{2}{8}$
• Add: $\frac{2}{8} + \frac{2}{8} = \frac{4}{8}$

Step 5: Simplify the result:

$\frac{4}{8}$ simplifies to $\frac{1}{2}$ by dividing the numerator and denominator by 4.

Therefore, the value of the expression is $\frac{1}{2}$.

Let's try to find the lowest common denominator between 3, 15, and 5

To find the lowest common denominator, we need to find a number that is divisible by 3, 15, and 5

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 5

We'll multiply the second fraction by 1

We'll multiply the third fraction by 3

$\frac{2\times5}{3\times5}+\frac{2\times1}{15\times1}-\frac{4\times3}{5\times3}=\frac{10}{15}+\frac{2}{15}-\frac{12}{15}$

Now we'll add and then subtract:

$\frac{10+2-12}{15}=\frac{12-12}{15}=\frac{0}{15}$

We'll divide both the numerator and denominator by 0 and get:

$\frac{0}{15}=0$

Let's try to find the lowest common denominator between 3, 15, and 5

To find the lowest common denominator, we need to find a number that is divisible by 3, 15, and 5

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 5

We'll multiply the second fraction by 1

We'll multiply the third fraction by 3

$\frac{1\times5}{3\times5}+\frac{7\times1}{15\times1}-\frac{2\times3}{5\times3}=\frac{5}{15}+\frac{7}{15}-\frac{6}{15}$

Now we'll add and then subtract:

$\frac{5+7-6}{15}=\frac{12-6}{15}=\frac{6}{15}$

We'll divide both numerator and denominator by 3 and get:

$\frac{6:3}{15:3}=\frac{2}{5}$

To solve the problem of adding the fractions $\frac{3}{7} + \frac{5}{14} + \frac{1}{3}$, we follow these steps:

• Step 1: Find the Least Common Denominator (LCD).
  We have denominators 7, 14, and 3. The least common multiple (LCM) of these numbers is 42.
• Step 2: Convert each fraction to the equivalent fraction with denominator 42.
  
  • $\frac{3}{7}$: Multiply both the numerator and the denominator by 6 to get $\frac{18}{42}$.
  • $\frac{5}{14}$: Multiply both the numerator and the denominator by 3 to get $\frac{15}{42}$.
  • $\frac{1}{3}$: Multiply both the numerator and the denominator by 14 to get $\frac{14}{42}$.
• Step 3: Add the fractions.
  Now, we add the numerators of these fractions: $18 + 15 + 14 = 47$.
• Final Result:
  The sum of the fractions is $\frac{47}{42}$.

Therefore, the final answer is $\frac{47}{42}$.

To solve this problem, we will add the fractions by finding a common denominator:

• Step 1: Identify the denominators: 2, 4, and 5. Find the least common multiple (LCM) of these denominators.
• Step 2: The LCM of 2, 4, and 5 is 20. Use this as the common denominator.
• Step 3: Convert each fraction to have a denominator of 20:
• Convert $\frac{1}{2}$ to $\frac{10}{20}$ by multiplying the numerator and denominator by 10.
• Convert $\frac{3}{4}$ to $\frac{15}{20}$ by multiplying the numerator and denominator by 5.
• Convert $\frac{2}{5}$ to $\frac{8}{20}$ by multiplying the numerator and denominator by 4.
• Step 4: Add the three fractions: $\frac{10}{20} + \frac{15}{20} + \frac{8}{20}$.
• Step 5: Add the numerators: $10 + 15 + 8 = 33$.
• Step 6: Write the result as a single fraction: $\frac{33}{20}$.
• Step 7: Check if the fraction can be simplified. Since 33 and 20 have no common factors other than 1, $\frac{33}{20}$ is in its simplest form.
• Step 8: Confirm this matches choice (4): $\frac{33}{20}$.

Therefore, the solution to the problem is $\frac{33}{20}$.

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

• Step 1: Determine the least common denominator (LCD) for all fractions.
• Step 2: Convert each fraction to have the LCD as the denominator.
• Step 3: Add the numerators of the fractions with the common denominator.
• Step 4: Simplify the resulting fraction if possible.

Now, let’s work through each step:

Step 1: The denominators are 3, 4, 6, and 12. The least common multiple of these numbers is 12. Thus, the LCD is 12.

Step 2: Convert each fraction:
- $\frac{2}{3}$ to an equivalent fraction with a denominator of 12:
Multiply both the numerator and denominator by 4 to get $\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$.
- $\frac{1}{4}$ to an equivalent fraction with a denominator of 12:
Multiply both the numerator and denominator by 3 to get $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.
- $\frac{5}{6}$ to an equivalent fraction with a denominator of 12:
Multiply both the numerator and denominator by 2 to get $\frac{5 \times 2}{6 \times 2} = \frac{10}{12}$.
- $\frac{1}{12}$ is already with the denominator 12, so it remains $\frac{1}{12}$.

Step 3: Add the numerators of these fractions:
$\frac{8}{12} + \frac{3}{12} + \frac{10}{12} + \frac{1}{12} = \frac{8 + 3 + 10 + 1}{12} = \frac{22}{12}$

Step 4: Simplify the fraction.
Since 22 and 12 share the common divisor 2, we simplify $\frac{22}{12}$ to $\frac{11}{6}$. This fraction cannot be simplified further.

Therefore, the solution to the problem is $\frac{11}{6}$.