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

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, 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 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 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 the problem of adding the two fractions $\frac{3}{8}$ and $\frac{1}{9}$, follow these steps:

• Step 1: Find the least common denominator (LCD) of the fractions. We calculate $8 \times 9 = 72$. Thus, the LCD is 72.
• Step 2: Convert each fraction to an equivalent fraction with the LCD as the new denominator.
• For $\frac{3}{8}$, multiply the numerator and denominator by 9: $\frac{3 \times 9}{8 \times 9} = \frac{27}{72}$.
• For $\frac{1}{9}$, multiply the numerator and denominator by 8: $\frac{1 \times 8}{9 \times 8} = \frac{8}{72}$.
• Step 3: Add the two fractions: $\frac{27}{72} + \frac{8}{72} = \frac{27 + 8}{72} = \frac{35}{72}$.
• Step 4: Simplify the resulting fraction if possible. Here, $\frac{35}{72}$ is already in its simplest form.

Thus, the sum of $\frac{3}{8}$ and $\frac{1}{9}$ is $\frac{35}{72}$.

Therefore, the solution to the problem is $\frac{35}{72}$.

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

• Step 1: Identify the common denominator.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Add the converted fractions.
• Step 4: Simplify the result.

Now, let's work through each step:
Step 1: The denominators are 7 and 8. Their product is $7 \times 8 = 56$. So, the common denominator is 56.
Step 2: Convert $\frac{1}{7}$ to have a denominator of 56 by multiplying numerator and denominator by 8: $\frac{1 \times 8}{7 \times 8} = \frac{8}{56}$.
Convert $\frac{1}{8}$ to have a denominator of 56 by multiplying numerator and denominator by 7: $\frac{1 \times 7}{8 \times 7} = \frac{7}{56}$.
Step 3: Add these equivalent fractions: $\frac{8}{56} + \frac{7}{56} = \frac{8 + 7}{56} = \frac{15}{56}$.
Step 4: The fraction $\frac{15}{56}$ is already in its simplest form.
Therefore, the solution to the problem is $\frac{15}{56}$.

To solve the problem of adding $\frac{2}{5}$ and $\frac{1}{6}$, we need to find a common denominator. We do this by multiplying the denominators: $5 \times 6 = 30$. This is the smallest common multiple of the two denominators and ensures that each fraction can be represented with a common base, allowing addition.

Let's convert each fraction to an equivalent fraction with the common denominator of 30:

• Convert $\frac{2}{5}$: Multiply both the numerator and the denominator by 6 to get $\frac{2 \times 6}{5 \times 6} = \frac{12}{30}$.

• Convert $\frac{1}{6}$: Multiply both the numerator and the denominator by 5 to get $\frac{1 \times 5}{6 \times 5} = \frac{5}{30}$.

Now, we add these equivalent fractions:

$\frac{12}{30} + \frac{5}{30} = \frac{12 + 5}{30} = \frac{17}{30}$.

The resulting fraction, $\frac{17}{30}$, is already in its simplest form because 17 is a prime number and does not share any common factors with 30 other than 1.

Thus, the sum of $\frac{2}{5}$ and $\frac{1}{6}$ is $\frac{17}{30}$.

Upon reviewing the given choices, the correct and matching choice is:

Choice 2: $\frac{17}{30}$

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 problem of adding $\frac{4}{15}$ and $\frac{1}{2}$, follow these steps:

• Step 1: Identify the denominators of the given fractions, which are $15$ and $2$.

• Step 2: Find the common denominator by multiplying the denominators: $15 \times 2 = 30$.

• Step 3: Adjust each fraction to have the common denominator:

  • Convert $\frac{4}{15}$ to $\frac{4 \times 2}{15 \times 2} = \frac{8}{30}$.

  • Convert $\frac{1}{2}$ to $\frac{1 \times 15}{2 \times 15} = \frac{15}{30}$.

• Step 4: Add the adjusted fractions:
  $\frac{8}{30} + \frac{15}{30} = \frac{8 + 15}{30} = \frac{23}{30}$.

• Step 5: Simplify the final expression. In this case, $\frac{23}{30}$ is already in simplest form.

The solution to the problem is $\frac{23}{30}$, which corresponds with choice 1 in the provided answer choices.

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 $\frac{15}{22}$.

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 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 the problem of adding $\frac{1}{4} + \frac{1}{3}$, we need to find a common denominator.

• Step 1: Determine the least common multiple (LCM) of the denominators. For 4 and 3, the LCM is 12.
• Step 2: Convert each fraction to an equivalent fraction with the denominator of 12.
  For $\frac{1}{4}$, multiply both numerator and denominator by 3: $\frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}$.
  For $\frac{1}{3}$, multiply both numerator and denominator by 4: $\frac{1 \cdot 4}{3 \cdot 4} = \frac{4}{12}$.
• Step 3: Add the resulting fractions: $\frac{3}{12} + \frac{4}{12} = \frac{3 + 4}{12} = \frac{7}{12}$.

Thus, the sum of $\frac{1}{4}$ and $\frac{1}{3}$ is $\frac{7}{12}$.

Therefore, the correct solution to the problem is $\frac{7}{12}$.

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 $\frac{13}{20}$.

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 $\frac{5}{6}$.

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 of $\frac{1}{6} + \frac{3}{7}$, we will use the following steps:

• Step 1: Find the common denominator for 6 and 7 by multiplying them. The common denominator is $6 \times 7 = 42$.
• Step 2: Express each fraction with this common denominator:
  • $\frac{1}{6} = \frac{1 \times 7}{6 \times 7} = \frac{7}{42}$
  • $\frac{3}{7} = \frac{3 \times 6}{7 \times 6} = \frac{18}{42}$
• Step 3: Add the adjusted fractions:
  $\frac{7}{42} + \frac{18}{42} = \frac{7 + 18}{42} = \frac{25}{42}$
• Step 4: Check if the fraction can be simplified further. In this case, $\frac{25}{42}$ is already in its simplest form.

The sum of $\frac{1}{6} + \frac{3}{7}$ is $\frac{25}{42}$.

The correct answer is choice 4: $\frac{25}{42}$.

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

• Step 1: Find a common denominator. The denominators are 5 and 4. The least common denominator is 20, which is the product of 5 and 4.
• Step 2: Convert each fraction to have the common denominator of 20.
  • For $\frac{3}{5}$, multiply both the numerator and the denominator by 4: $\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$.
  • For $\frac{1}{4}$, multiply both the numerator and denominator by 5: $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$.
• Step 3: Add the equivalent fractions: $\frac{12}{20} + \frac{5}{20} = \frac{12 + 5}{20} = \frac{17}{20}$.

Thus, the sum of $\frac{3}{5}$ and $\frac{1}{4}$ is $\frac{17}{20}$.

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.