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

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 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 $\frac{1}{2}$ 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 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 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 $\frac{11}{14}$.

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 $\frac{1}{4}$ 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 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 $\frac{2}{8}$ 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 problem $\frac{2}{5} + \frac{2}{6}$, we need a common denominator:

• Step 1: Find the least common multiple (LCM) of the denominators 5 and 6. The LCM of 5 and 6 is 30.
• Step 2: Rewrite each fraction to have the denominator 30:
  • For $\frac{2}{5}$: Multiply the numerator and denominator by $6$ (since $5 \times 6 = 30$). This gives $\frac{2 \times 6}{5 \times 6} = \frac{12}{30}$.
  • For $\frac{2}{6}$: Multiply the numerator and denominator by $5$ (since $6 \times 5 = 30$). This gives $\frac{2 \times 5}{6 \times 5} = \frac{10}{30}$.
• Step 3: Add the two fractions: $\frac{12}{30} + \frac{10}{30} = \frac{22}{30}$.
• Step 4: Simplify the resulting fraction. Since $\frac{22}{30}$ is already in simplest form, no further simplification is necessary.

Therefore, the sum of $\frac{2}{5}$ and $\frac{2}{6}$ is $\frac{22}{30}$, which corresponds to choice 4.

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

• Step 1: Find a common denominator for the fractions. The denominators are $7$ and $3$. By multiplying $7$ and $3$, we find the common denominator is $21$.
• Step 2: Convert each fraction to an equivalent fraction with this common denominator.
  • The first fraction: $\frac{3}{7}$. We multiply both the numerator and the denominator by $3$ (since $7 \times 3 = 21$): $\frac{3 \times 3}{7 \times 3} = \frac{9}{21}.$
  • The second fraction: $\frac{1}{3}$. We multiply both the numerator and the denominator by $7$ (since $3 \times 7 = 21$): $\frac{1 \times 7}{3 \times 7} = \frac{7}{21}.$
• Step 3: Add the fractions. Now that both fractions have the same denominator, we can add them: $\frac{9}{21} + \frac{7}{21} = \frac{9 + 7}{21} = \frac{16}{21}.$
• Step 4: Simplify the resulting fraction if necessary. In this case, $\frac{16}{21}$ is already in its simplest form.

Therefore, the final solution to the problem is $\frac{16}{21}$.

To solve the problem of adding the fractions $\frac{1}{4}$ and $\frac{3}{9}$, we will first find a common denominator and then perform the addition:

Step 1: Finding a Common Denominator
The denominators are 4 and 9. The easiest way to find a common denominator is to multiply these two numbers. Hence, $4 \times 9 = 36$ gives us a common denominator of 36.

Step 2: Convert Each Fraction
Convert $\frac{1}{4}$ to a fraction with denominator 36. To do this, multiply the numerator and denominator by 9 (since $4 \times 9 = 36$):
$\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}$

Next, convert $\frac{3}{9}$ to a fraction with denominator 36. Multiply the numerator and denominator by 4 (since $9 \times 4 = 36$):
$\frac{3}{9} = \frac{3 \times 4}{9 \times 4} = \frac{12}{36}$

Step 3: Add the Fractions
Now add the two fractions:
$\frac{9}{36} + \frac{12}{36} = \frac{9+12}{36} = \frac{21}{36}$

Step 4: Simplify the Result (if necessary)
The fraction $\frac{21}{36}$ can be simplified by finding the greatest common divisor (GCD) of 21 and 36, which is 3. However, in the current situation with the answer choices provided, $\frac{21}{36}$ matches one of the options directly without further simplification, ensuring it meets the expected answer format.

Therefore, the sum of $\frac{1}{4} + \frac{3}{9}$ is $\frac{21}{36}$, which corresponds to choice $1$.

Thus, the correct answer is $\frac{21}{36}$.

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 $\frac{13}{15}$.

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 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 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 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: $\frac{17}{30}$