All Operations in Fractions: The common denominator is smaller than the product of the denominators

We need to find a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{6}$ in order to add them together.

Step 1: Identify the least common denominator (LCD).

• The denominators are 3 and 6.
• The least common multiple (LCM) of 3 and 6 is 6. Hence, the LCD is 6.

Step 2: Convert each fraction to an equivalent fraction with the LCD of 6.

• $\frac{1}{3}$ needs to be converted. Multiply both numerator and denominator by 2: $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.
• $\frac{1}{6}$ already has the denominator as 6, so it remains $\frac{1}{6}$.

Step 3: Add the fractions.

• Now that the denominators are the same, we can add the numerators: $\frac{2}{6} + \frac{1}{6} = \frac{2 + 1}{6} = \frac{3}{6}$.

Step 4: Simplify the result.

• $\frac{3}{6}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3: $\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$.

Thus, the result of the addition of $\frac{1}{3}$ and $\frac{1}{6}$ is $\frac{1}{2}$.

Therefore, the solution to the problem is $\frac{1}{2}$.

To solve the problem of adding the fractions $\frac{3}{4}$ and $\frac{1}{6}$, we need to find a common denominator.

• Step 1: Find the LCM of the denominators:
  The denominators are 4 and 6. The LCM of 4 and 6 is 12.
• Step 2: Convert each fraction to have the common denominator:
  - Convert $\frac{3}{4}$ to an equivalent fraction with a denominator of 12. To do this, multiply both the numerator and the denominator by 3:
  $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.
  - Convert $\frac{1}{6}$ to an equivalent fraction with a denominator of 12. Multiply both the numerator and the denominator by 2:
  $\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$.
• Step 3: Add the fractions:
  Now that both fractions have the same denominator, add the numerators:
  $\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$.
• Step 4: Simplify if necessary:
  The fraction $\frac{11}{12}$ is already in its simplest form.

Therefore, the solution to the problem $\frac{3}{4} + \frac{1}{6}$ is $\frac{11}{12}$.

To solve the problem of adding the fractions $\frac{1}{2}$ and $\frac{4}{6}$, we start by finding the least common denominator (LCD).

First, we identify the denominators: 2 and 6. The least common multiple of 2 and 6 is 6, which will be our LCD.

Next, we convert each fraction to have the denominator of 6:

• Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 6. Since $2 \cdot 3 = 6$, multiply the numerator by 3: $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.

• The fraction $\frac{4}{6}$ already has the desired common denominator.

Now that the fractions are $\frac{3}{6}$ and $\frac{4}{6}$, we can add them:

$\frac{3}{6} + \frac{4}{6} = \frac{3+4}{6} = \frac{7}{6}$.

The solution to the problem is $\frac{7}{6}$, which matches choice 2.

To find the sum $\frac{1}{4} + \frac{7}{8}$, follow these steps:

• Step 1: Identify the least common denominator (LCD) of the fractions. The denominators 4 and 8 have an LCD of 8.
• Step 2: Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of 8. Multiply both the numerator and the denominator by 2: $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$.
• Step 3: The second fraction, $\frac{7}{8}$, already has the correct denominator. Therefore, it remains $\frac{7}{8}$.
• Step 4: Add the numerators of the two fractions: $\frac{2}{8} + \frac{7}{8} = \frac{2+7}{8} = \frac{9}{8}$.

Therefore, the sum of $\frac{1}{4}$ and $\frac{7}{8}$ is $\frac{9}{8}$.

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

• Step 1: Identify the denominators of the fractions.
• Step 2: Because the denominators are the same, add the numerators.
• Step 3: Simplify the resulting fraction if necessary.

Now, let's work through each step:
Step 1: Both fractions, $\frac{1}{4}$ and $\frac{3}{4}$, have the same denominator, 4.
Step 2: Since the denominators are the same, we can add the numerators: $1 + 3 = 4$.
Step 3: The resulting fraction is $\frac{4}{4}$, which simplifies to $1$.

Therefore, the solution to the problem is $1$.

To solve the problem of adding $\frac{1}{2}$ and $\frac{1}{6}$, we need to follow these steps:

• Step 1: Determine the least common denominator (LCD).
• Step 2: Convert the fractions to have this common denominator.
• Step 3: Add the fractions.
• Step 4: Simplify the result if necessary.

Step 1: The denominators are 2 and 6. The least common multiple of 2 and 6 is 6.

Step 2: We convert each fraction:
- Convert $\frac{1}{2}$ to a denominator of 6: $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.
- The fraction $\frac{1}{6}$ already has the denominator 6.

Step 3: Add the fractions with common denominators:
$\frac{3}{6} + \frac{1}{6} = \frac{3 + 1}{6} = \frac{4}{6}.$

Step 4: Simplify the fraction $\frac{4}{6}$.
The greatest common divisor of 4 and 6 is 2, so divide both the numerator and the denominator by 2:
$\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}.$

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

To solve the addition of the fractions $\frac{4}{6} + \frac{1}{8}$, we will first find the least common denominator.

• The denominators of the fractions are 6 and 8. To add these fractions, we need a common denominator.
• Calculate the least common multiple (LCM) of 6 and 8:
• Prime factorization of 6: $6 = 2 \times 3$.
• Prime factorization of 8: $8 = 2^3$.
• The LCM will take the highest power of each prime that appears in these factorizations: $2^3 \times 3 = 24$.
• Convert each fraction to an equivalent fraction with 24 as the denominator:
• Convert $\frac{4}{6}$: Multiply both the numerator and denominator by 4 (since $\frac{24}{6} = 4$): $\frac{4 \times 4}{6 \times 4} = \frac{16}{24}$.
• Convert $\frac{1}{8}$: Multiply both the numerator and denominator by 3 (since $\frac{24}{8} = 3$): $\frac{1 \times 3}{8 \times 3} = \frac{3}{24}$.
• Now, add these two fractions:
• $\frac{16}{24} + \frac{3}{24} = \frac{16 + 3}{24} = \frac{19}{24}$.

Thus, the sum of the fractions $\frac{4}{6}$ and $\frac{1}{8}$ is $\frac{19}{24}$.

The correct choice from the available options is $\frac{19}{24}$.

Therefore, the solution to the problem is $\frac{19}{24}$.

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

• Step 1: Determine the least common denominator of 4 and 6.
• Step 2: Convert each fraction to this common denominator.
• Step 3: Add the numerators and form the resultant fraction.
• Step 4: Simplify the resulting fraction if necessary.

Now, let's work through each step:
Step 1: The denominators are 4 and 6. The least common multiple of 4 and 6 is 12.

Step 2: Convert each fraction to have the denominator 12.
For $\frac{1}{4}$, multiplying the numerator and denominator by 3 gives $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.
For $\frac{2}{6}$, multiplying the numerator and denominator by 2 gives $\frac{2 \times 2}{6 \times 2} = \frac{4}{12}$.

Step 3: Add the fractions: $\frac{3}{12} + \frac{4}{12} = \frac{3 + 4}{12} = \frac{7}{12}$.

Step 4: Check if $\frac{7}{12}$ can be simplified. Since 7 and 12 have no common factors other than 1, it is already in its simplest form.

Therefore, the sum of $\frac{1}{4} + \frac{2}{6}$ is $\frac{7}{12}$.

To solve the fraction addition problem $\frac{2}{4} + \frac{2}{6}$, follow these steps:

• Step 1: Identify the least common denominator (LCD) of the fractions. The denominators are 4 and 6. The factors of 4 are 2 and 2, and the factors of 6 are 2 and 3. The LCD is the smallest number that both denominators divide into, which is 12.

• Step 2: Convert each fraction to an equivalent fraction with the denominator of 12.

• Step 3: For $\frac{2}{4}$:

  • Find the equivalent fraction: Multiply both the numerator and denominator by 3 (since 4 * 3 = 12).

  • The equivalent fraction is $\frac{2 \times 3}{4 \times 3} = \frac{6}{12}$.

• Step 4: For $\frac{2}{6}$:

  • Find the equivalent fraction: Multiply both the numerator and denominator by 2 (since 6 * 2 = 12).

  • The equivalent fraction is $\frac{2 \times 2}{6 \times 2} = \frac{4}{12}$.

• Step 5: Add the new fractions: $\frac{6}{12} + \frac{4}{12} = \frac{10}{12}$.

Therefore, the sum of the fractions is $\boxed{\frac{10}{12}}$.

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

• Step 1: Find the Least Common Denominator (LCD):
  The denominators are 4 and 6. The least common multiple of 4 and 6 is 12.
• Step 2: Convert Each Fraction:
  - Convert $\frac{1}{4}$ to a fraction with a denominator of 12:
  $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
  - Convert $\frac{4}{6}$ to a fraction with a denominator of 12:
  $\frac{4}{6} = \frac{4 \times 2}{6 \times 2} = \frac{8}{12}$
• Step 3: Add the Fractions:
  Now, add the fractions: $\frac{3}{12} + \frac{8}{12} = \frac{3 + 8}{12} = \frac{11}{12}$
• Step 4: Simplify the Fraction (if needed):
  The fraction $\frac{11}{12}$ is already in its simplest form.

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

To solve the addition of these two fractions, we'll proceed as follows:

• Step 1: Determine the least common denominator (LCD) of the fractions.
  The denominators are 4 and 6, and the smallest number that is a multiple of both is 12. Thus, the LCD is 12.
• Step 2: Convert each fraction to an equivalent fraction with the denominator 12.
  - For $\frac{3}{4}$, multiply both numerator and denominator by 3: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.
  - For $\frac{1}{6}$, multiply both numerator and denominator by 2: $\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$.
• Step 3: Add the converted fractions.
  $\frac{9}{12} + \frac{2}{12} = \frac{9 + 2}{12} = \frac{11}{12}$.

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

To solve the problem of adding $\frac{2}{4}$ and $\frac{1}{6}$, follow these steps:

Step 1: Identify the least common denominator of the fractions.

The denominators of the fractions are 4 and 6. The least common multiple of 4 and 6 is 12, so 12 is our common denominator.

Step 2: Convert each fraction to an equivalent fraction with the denominator of 12.

• For $\frac{2}{4}$: Multiply both numerator and denominator by 3 to obtain $\frac{6}{12}$. This is because $4 \times 3 = 12$.

• For $\frac{1}{6}$: Multiply both numerator and denominator by 2 to obtain $\frac{2}{12}$. This is because ${6 \times 2 = 12}$.

Step 3: Add the converted fractions.

$\frac{6}{12} + \frac{2}{12} = \frac{6 + 2}{12} = \frac{8}{12}$

Step 4: Simplify the final fraction if possible.

In this case, $\frac{8}{12}$ can be simplified by dividing numerator and denominator by their greatest common divisor, which is 4. Thus, $\frac{8}{12}$ simplifies to $\frac{2}{3}$.

However, as per the problem's required answer, the unsimplified fraction is $\frac{8}{12}$.

Therefore, the solution to the problem is:

$\frac{8}{12}$

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

• Step 1: Identify the given fractions, $\frac{3}{5}$ and $\frac{3}{10}$.
• Step 2: Find the least common denominator (LCD) of the denominators 5 and 10.
• Step 3: Convert each fraction to have this common denominator.
• Step 4: Add the fractions.

Now, let's work through each step:

Step 1: We have the fractions $\frac{3}{5}$ and $\frac{3}{10}$.

Step 2: Check the denominators. The denominators are 5 and 10. The least common multiple of 5 and 10 is 10. Thus, our common denominator will be 10.

Step 3: Convert each fraction to an equivalent fraction with a denominator of 10:
$\frac{3}{5}$ is equivalent to $\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$ (since $\frac{5}{10}$ must equal 10, multiply both numerator and denominator by 2).

Step 4: The fraction $\frac{3}{10}$ already has the denominator of 10.
Thus, $\frac{3}{5} + \frac{3}{10} = \frac{6}{10} + \frac{3}{10} = \frac{6 + 3}{10} = \frac{9}{10}$.

Therefore, the answer to the problem is $\frac{9}{10}$, which corresponds to choice 1.

To solve this problem, we will perform the following steps:

• Step 1: Find the Least Common Multiple (LCM) of the denominators 3 and 9.
• Step 2: Convert both fractions to have a common denominator.
• Step 3: Add the numerators of the converted fractions and write the result over the common denominator.
• Step 4: Simplify the resultant fraction if needed.

Let's work through each step:

Step 1:
The denominators of our fractions are 3 and 9. The LCM of 3 and 9 is 9, since 9 is the smallest number that both 3 and 9 divide evenly into.

Step 2:
Convert each fraction to have a denominator of 9.

- For $\frac{2}{3}$, multiply both the numerator and denominator by 3 (because $\frac{9}{3} = 3$):
$\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}$

- The second fraction $\frac{7}{9}$ already has a denominator of 9, so it remains the same:
$\frac{7}{9}$

Step 3:
Add the two fractions:
$\frac{6}{9} + \frac{7}{9} = \frac{6 + 7}{9} = \frac{13}{9}$

Step 4:
The fraction $\frac{13}{9}$ is in its simplest form because 13 is a prime number and does not divide evenly into 9.

Therefore, the solution to the problem is $\frac{13}{9}$.

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

• Step 1: Identify the Least Common Denominator (LCD) for the two fractions.
• Step 2: Convert each fraction to have the LCD as the denominator.
• Step 3: Add the fractions with the common denominator.

Now, let's work through each step:

Step 1: The denominators of the fractions are 10 and 5. The LCD of 10 and 5 is 10.
Step 2: Convert $\frac{2}{5}$ to have a denominator of 10. We can multiply both the numerator and the denominator by 2: $\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$.
Step 3: Now, add $\frac{9}{10}$ and $\frac{4}{10}$ (since both fractions now have the same denominator): $\frac{9}{10} + \frac{4}{10} = \frac{9 + 4}{10} = \frac{13}{10}$.

The two fractions added together give us $\frac{13}{10}$. Therefore, the solution to the problem is $\frac{13}{10}$, which matches the correct answer choice.

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

• Step 1: Identify the least common multiple (LCM) of the denominators 8 and 4.
• Step 2: Convert $\frac{1}{4}$ to a fraction with the denominator of 8.
• Step 3: Add the fractions $\frac{3}{8}$ and $\frac{2}{8}$.

Now, let's work through each step:
Step 1: The LCM of 8 and 4 is 8, so this will be the common denominator.
Step 2: Transform $\frac{1}{4}$ into a fraction with the denominator of 8. Multiply the numerator and the denominator by 2 to get $\frac{2}{8}$.
Step 3: Now, add the fractions with the same denominator: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.

Therefore, the sum of $\frac{3}{8}$ and $\frac{1}{4}$ is $\frac{5}{8}$, which matches choice 4.

To solve the problem $\frac{4}{15} + \frac{2}{5}$, we will add these fractions by finding a common denominator.

Step 1: Find the Least Common Denominator (LCD).
The denominators are 15 and 5. The LCM of 15 and 5 is 15, as 15 is already a multiple of 5.

Step 2: Convert each fraction to have the same denominator, 15.
- The fraction $\frac{4}{15}$ already has the denominator 15. - Convert $\frac{2}{5}$ to a fraction with a denominator of 15 by multiplying both the numerator and denominator by 3: $\frac{2 \times 3}{5 \times 3} = \frac{6}{15}$.

Step 3: Add the fractions with common denominators.
Now we have: $\frac{4}{15} + \frac{6}{15}$.
Add the numerators: $4 + 6 = 10$.
The new fraction is $\frac{10}{15}$.

Step 4: Simplify the resulting fraction, if possible.
Both 10 and 15 are divisible by 5.
Divide the numerator and the denominator by their greatest common divisor (GCD), which is 5: $\frac{10 \div 5}{15 \div 5} = \frac{2}{3}$.

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

To solve the problem of adding $\frac{3}{14} + \frac{3}{7}$, we need the following steps:

• Step 1: Identify the least common denominator of $14$ and $7$. Since $7$ is a factor of $14$, the least common denominator is $14$.
• Step 2: Rewrite the fractions with the common denominator.
  The fraction $\frac{3}{7}$ can be converted to an equivalent fraction with the denominator $14$:
• Multiply the numerator and the denominator of $\frac{3}{7}$ by $2$ (since $7 \times 2 = 14$) to get $\frac{6}{14}$.
• Step 3: Add $\frac{3}{14}$ and $\frac{6}{14}$ now that they have the same denominator:
  $\frac{3}{14} + \frac{6}{14} = \frac{3+6}{14} = \frac{9}{14}$.
• Step 4: Simplify if necessary. The numerator and denominator here are coprime, so $\frac{9}{14}$ is already in its simplest form.

Thus, the sum of the fractions is $\frac{9}{14}$.

To solve the problem of adding $\frac{3}{4}$ and $\frac{3}{8}$, let's follow a systematic approach:

• Step 1: Identify the Common Denominator
  The denominators are 4 and 8. The least common multiple (LCM) of 4 and 8 is 8. Thus, 8 will be our common denominator.
• Step 2: Convert Fractions to Common Denominator
  The fraction $\frac{3}{4}$ needs to be converted to an equivalent fraction with a denominator of 8. To do this, multiply both the numerator and the denominator by 2: $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$.
  The fraction $\frac{3}{8}$ already has a denominator of 8, so it remains the same: $\frac{3}{8}$.
• Step 3: Add the Fractions
  With a common denominator, add the numerators while keeping the denominator the same: $\frac{6}{8} + \frac{3}{8} = \frac{6 + 3}{8} = \frac{9}{8}$.
• Step 4: Simplify the Fraction if Necessary
  The fraction $\frac{9}{8}$ is already in its simplest form, but it is an improper fraction. If desired, it can be expressed as a mixed number: $1 \frac{1}{8}$. However, $\frac{9}{8}$ as a fraction suffices for this problem.

Therefore, the solution to the problem is $\frac{9}{8}$.

To solve this problem, we will add the fractions $\frac{4}{12}$ and $\frac{5}{6}$. Follow these steps:

• Step 1: Find the least common denominator (LCD) of the fractions. The denominators are 12 and 6. The LCD is 12 because it is the smallest number into which both 12 and 6 divide evenly.
• Step 2: Convert the fractions to have the common denominator 12. The fraction $\frac{4}{12}$ is already expressed with this denominator.
• Step 3: Convert $\frac{5}{6}$ to an equivalent fraction with a denominator of 12. To do this, determine what number times 6 gives 12, which is 2. Multiply both the numerator and the denominator of $\frac{5}{6}$ by 2:
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
• Step 4: Add the fractions now that they have the same denominator:
$\frac{4}{12} + \frac{10}{12} = \frac{4 + 10}{12} = \frac{14}{12}$
• Step 5: Simplify the resulting fraction, $\frac{14}{12}$. Both the numerator and the denominator are divisible by 2, so:
$\frac{14}{12} = \frac{14 \div 2}{12 \div 2} = \frac{7}{6}$

Thus, the sum of the fractions $\frac{4}{12}$ and $\frac{5}{6}$ is $\frac{7}{6}$.

Therefore, the correct answer is $\frac{7}{6}$.