All Operations in Fractions: Finding a Common Denominator by Multiplying the Denominators

To solve $\frac{1}{2} - \frac{1}{9}$, follow these steps:

Step 1: Find the least common multiple (LCM) of the denominators 2 and 9.
The multiples of 2 are $2, 4, 6, 8, 10, 12, 14, 16, 18, \ldots$
The multiples of 9 are $9, 18, 27, \ldots$
The smallest common multiple is 18. Thus, the LCM of 2 and 9 is 18.

Step 2: Convert each fraction to an equivalent fraction with the common denominator 18.
For $\frac{1}{2}$, the equivalent fraction with 18 as the denominator is calculated by finding the factor needed:
$\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}$.
For $\frac{1}{9}$, the equivalent fraction with 18 as the denominator is:
$\frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18}$.

Step 3: Perform the subtraction of these equivalent fractions.
$\frac{9}{18} - \frac{2}{18} = \frac{9 - 2}{18} = \frac{7}{18}$.

Therefore, the solution to the problem is $\boxed{\frac{7}{18}}$.

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 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 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, 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 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 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 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 $\frac{2}{5}$ 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 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 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 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 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 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 problem of subtracting $\frac{1}{4}$ from $\frac{3}{5}$, we need a common denominator.

First, find the least common denominator (LCD) of 5 and 4, which is 20. This is done by multiplying the denominators: $5 \times 4 = 20$.

Next, convert each fraction to an equivalent fraction with the denominator of 20:

• For $\frac{3}{5}$: Multiply both numerator and denominator by 4 to get $\frac{3 \times 4}{5 \times 4} = \frac{12}{20}$.
• For $\frac{1}{4}$: Multiply both numerator and denominator by 5 to get $\frac{1 \times 5}{4 \times 5} = \frac{5}{20}$.

Now perform the subtraction with these equivalent fractions:

$\frac{12}{20} - \frac{5}{20} = \frac{12 - 5}{20} = \frac{7}{20}$

The resulting fraction, $\frac{7}{20}$, is already in its simplest form.

Therefore, the solution to the subtraction $\frac{3}{5} - \frac{1}{4}$ is $\frac{7}{20}$.

Checking against the multiple-choice answers, the correct choice is the first one: $\frac{7}{20}$.

To solve the problem $\frac{1}{3} - \frac{1}{5}$, we follow these steps:

First, we need to find a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{5}$. The denominators are 3 and 5, and their least common multiple (LCM) is 15.

We will convert each fraction to an equivalent fraction with the denominator 15:

• To convert $\frac{1}{3}$ to a fraction with denominator 15, multiply both the numerator and the denominator by 5: $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$
• To convert $\frac{1}{5}$ to a fraction with denominator 15, multiply both the numerator and the denominator by 3: $\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

Now that both fractions have the same denominator, we can subtract the numerators:

Therefore, the solution to the problem is $\frac{2}{15}$.

To solve the subtraction of fractions $\frac{3}{5} - \frac{1}{2}$, we will follow these steps:

• Step 1: Find the least common multiple (LCM) of the denominators 5 and 2. The LCM of 5 and 2 is 10.
• Step 2: Convert each fraction to have a denominator of 10.
• Step 3: Subtract the converted fractions.
• Step 4: Simplify the result if necessary.

Now, let's work through each step in detail:

Step 1: The LCM of 5 and 2 is 10, since 10 is the smallest number that both 5 and 2 divide into evenly.

Step 2: Convert each fraction to have a denominator of 10.

For $\frac{3}{5}$:
Multiply numerator and denominator by 2 to get $\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$.

For $\frac{1}{2}$:
Multiply numerator and denominator by 5 to get $\frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

Step 3: Subtract the fractions:

$\frac{6}{10} - \frac{5}{10} = \frac{6 - 5}{10} = \frac{1}{10}$.

Step 4: There is no further simplification needed for $\frac{1}{10}$ as it is already in its simplest form.

Therefore, the solution to the problem is $\frac{1}{10}$.

The correct answer, choice (4), is $\frac{1}{10}$.

To solve the subtraction of fractions $\frac{3}{5} - \frac{1}{3}$, follow these steps:

• Step 1: Find the Least Common Denominator (LCD)
  The denominators are 5 and 3. The least common multiple of 5 and 3 is 15. Thus, the common denominator will be 15.
• Step 2: Convert fractions to have the same denominator
  For $\frac{3}{5}$, multiply both the numerator and the denominator by 3 to get:
  $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$.
  For $\frac{1}{3}$, multiply both the numerator and the denominator by 5 to get:
  $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Subtract the numerators
  Now subtract the equivalent fractions:
  $\frac{9}{15} - \frac{5}{15} = \frac{9 - 5}{15} = \frac{4}{15}$.
• Step 4: Simplify the fraction
  The fraction $\frac{4}{15}$ is already in its simplest form.

Thus, the solution to the problem is $\frac{4}{15}$.

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 $\frac{5}{6}$.