Addition of Fractions: The common denominator is smaller than the product of the denominators

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 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 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 $\frac{1}{4} + \frac{4}{6}$, 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 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 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 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 addition of fractions $\frac{2}{6} + \frac{3}{9}$, we will follow these logical steps:

• Step 1: Find a common denominator.
  The denominators are $6$ and $9$. The least common multiple (LCM) of these numbers is $18$.
• Step 2: Convert each fraction to have the denominator of $18$.
  To convert $\frac{2}{6}$ to a denominator of $18$, multiply both the numerator and the denominator by $3$ (because $6 \times 3 = 18$): $\frac{2}{6} = \frac{2 \times 3}{6 \times 3} = \frac{6}{18}$ To convert $\frac{3}{9}$ to a denominator of $18$, multiply both the numerator and the denominator by $2$ (because $9 \times 2 = 18$): $\frac{3}{9} = \frac{3 \times 2}{9 \times 2} = \frac{6}{18}$
• Step 3: Add the fractions.
  Now that both fractions have the same denominator, add their numerators: $\frac{6}{18} + \frac{6}{18} = \frac{6 + 6}{18} = \frac{12}{18}$
• Step 4: Simplify if possible.
  Check if $\frac{12}{18}$ can be simplified. The greatest common divisor (GCD) of $12$ and $18$ is $6$, so: $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$ However, since the original question focused on achieving the fraction with denominator $18$, our final non-simplified answer remains $\frac{12}{18}$.

The final result is that the sum of the fractions is $\frac{12}{18}$.

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 $\frac{1}{2}$.

The problem involves adding the fractions $\frac{1}{3}$ and $\frac{4}{9}$.

Step 1: Identify the Least Common Denominator (LCD).

• The denominators are 3 and 9. The least common multiple of 3 and 9 is 9. Thus, the LCD is 9.

Step 2: Convert the fractions to have the common denominator.

• The fraction $\frac{1}{3}$ must be converted to have the denominator of 9. Multiply both the numerator and denominator by 3:
• $\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}$
• The fraction $\frac{4}{9}$ already has the denominator of 9, so it remains unchanged.

Step 3: Add the equivalent fractions.

• Add the numerators together, keeping the denominator:
• $\frac{3}{9} + \frac{4}{9} = \frac{3+4}{9} = \frac{7}{9}$

Step 4: Simplify the result, if necessary.

• The fraction $\frac{7}{9}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{7}{9}$.

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

• Step 1: Find the least common multiple (LCM) of the denominators 10 and 4.
• Step 2: Convert both fractions to have the common denominator.
• Step 3: Add the numerators and present the result.

Now, let's work through each step:

Step 1: Find the LCM of 10 and 4. The prime factors of 10 are $2 \times 5$, and for 4, $2^2$. The LCM is $2^2 \times 5 = 20$.

Step 2: Convert the fractions:
$\frac{5}{10}$ can be converted by multiplying both the numerator and the denominator by 2: $\frac{5 \times 2}{10 \times 2} = \frac{10}{20}$.
$\frac{1}{4}$ can be converted by multiplying both the numerator and the denominator by 5: $\frac{1 \times 5}{4 \times 5} = \frac{5}{20}$.

Step 3: Add the fractions:
$\frac{10}{20} + \frac{5}{20} = \frac{10 + 5}{20} = \frac{15}{20}$.

Therefore, the solution to the problem is $\frac{15}{20}$, which matches choice ID 4.

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

Firstly, we find a common denominator for the fractions. The denominators are 5 and 10. The least common multiple (LCM) of 5 and 10 is 10.

Next, we convert each fraction to an equivalent fraction with the denominator of 10:

• The fraction $\frac{2}{5}$ is equivalent to $\frac{2 \times 2}{5 \times 2} = \frac{4}{10}$.
• The fraction $\frac{3}{10}$ is already expressed with the denominator of 10, so it remains as $\frac{3}{10}$.

Now, we add both fractions: $\frac{4}{10} + \frac{3}{10} = \frac{4+3}{10} = \frac{7}{10}$.

Therefore, the solution to the exercise is $\frac{7}{10}$.

We must first identify the lowest common denominator between 6 and 9.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 6 and 9.

In this case, the common denominator is 18.

We will then proceed to multiply each fraction by the appropriate number to reach the denominator 18.

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{3\times3}{6\times3}+\frac{3\times2}{9\times2}=\frac{9}{18}+\frac{6}{18}$

Finally we'll combine and obtain the following:

$\frac{9+6}{18}=\frac{15}{18}$

Let's first identify the lowest common denominator between 8 and 12.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 8 and 12.

In this case, the common denominator is 24

Now we'll proceed to multiply each fraction by the appropriate number to reach the denominator 24.p

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{2\times3}{8\times3}+\frac{5\times2}{12\times2}=\frac{6}{24}+\frac{10}{24}$

Now let's combine:

$\frac{6+10}{24}=\frac{16}{24}$

To solve the problem of adding the fractions $\frac{1}{5}$ and $\frac{7}{10}$, we follow these steps:

• Step 1: Identify the least common multiple (LCM) of the denominators 5 and 10, which is 10.
• Step 2: Convert $\frac{1}{5}$ to a fraction with a denominator of 10. To do this, multiply both the numerator and denominator by 2: $\frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10}$
• Step 3: Observe that $\frac{7}{10}$ already has the common denominator of 10.
• Step 4: Add the two fractions with a common denominator: $\frac{2}{10} + \frac{7}{10} = \frac{2 + 7}{10} = \frac{9}{10}$

The sum of $\frac{1}{5}$ and $\frac{7}{10}$ is thus $\mathbf{\frac{9}{10}}$.

To solve the problem of adding $\frac{3}{8}$ and $\frac{5}{12}$, follow these steps:

• Step 1: Find the least common multiple (LCM) of the denominators 8 and 12. The LCM of 8 and 12 is 24.
• Step 2: Convert the fractions to have the common denominator 24.
  To convert $\frac{3}{8}$ to a denominator of 24:
  Multiply both the numerator and denominator by 3: $\frac{3 \times 3}{8 \times 3} = \frac{9}{24}$.
• Step 3: Convert $\frac{5}{12}$ to a denominator of 24:
  Multiply both the numerator and denominator by 2: $\frac{5 \times 2}{12 \times 2} = \frac{10}{24}$.
• Step 4: Add the fractions $\frac{9}{24} + \frac{10}{24}$.
  Since they share the same denominator, add the numerators: $9 + 10 = 19$.
• Step 5: The sum is $\frac{19}{24}$. There is no need to simplify further, as 19 and 24 have no common factors other than 1.

Thus, the fraction $\frac{3}{8} + \frac{5}{12}$ simplifies to $\frac{19}{24}$.

To solve this problem, we will follow these steps:

• Step 1: Find the least common denominator of 10 and 6.
• Step 2: Convert both fractions to have this common denominator.
• Step 3: Add the fractions.
• Step 4: Simplify the resulting fraction if possible.

Now, let's work through each step:

Step 1: The least common multiple (LCM) of 10 and 6 is 30. So, our common denominator will be 30.

Step 2: Convert each fraction to have a denominator of 30:
For $\frac{5}{10}$:
$\frac{5}{10} = \frac{5 \times 3}{10 \times 3} = \frac{15}{30}$
For $\frac{1}{6}$:
$\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}$

Step 3: Add the fractions:
$\frac{15}{30} + \frac{5}{30} = \frac{15 + 5}{30} = \frac{20}{30}$

Step 4: Simplify if needed:
The fraction $\frac{20}{30}$ is already simplified to one of the given answer choices with a common denominator, matching one of the options.

Therefore, the solution to the problem is $\frac{20}{30}$.

Let's first identify the lowest common denominator between 8 and 12.

In order to identify the lowest common denominator, we need to find a number that is divisible by both 8 and 12.

In this case, the common denominator is 24.

Let's proceed to multiply each fraction by the appropriate number to reach the denominator 24.

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{4\times3}{8\times3}+\frac{5\times2}{12\times2}=\frac{12}{24}+\frac{10}{24}$

Now let's add:

$\frac{12+10}{24}=\frac{22}{24}$

To solve for the sum of $\frac{4}{10} + \frac{2}{6}$, we will proceed with the following steps:

• Step 1: Identify the least common denominator (LCD)
• Step 2: Convert each fraction to an equivalent fraction with this common denominator
• Step 3: Add the numerators

Let's begin:

Step 1: Identify the least common denominator (LCD)
The denominators are 10 and 6. The least common multiple (LCM) of 10 and 6 can be found by evaluating their prime factors:
10 = 2 × 5
6 = 2 × 3
The LCM is found by taking the highest power of each prime that appears:
LCM = 2 × 3 × 5 = 30.
Thus, the common denominator is 30.

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 30
For $\frac{4}{10}$:
Multiply both the numerator and the denominator by 3 to make the denominator 30:
$\frac{4}{10} = \frac{4 \times 3}{10 \times 3} = \frac{12}{30}$.
For $\frac{2}{6}$:
Multiply both the numerator and the denominator by 5 to make the denominator 30:
$\frac{2}{6} = \frac{2 \times 5}{6 \times 5} = \frac{10}{30}$.

Step 3: Add the numerators
Now that the fractions have the same denominator, add the numerators:
$\frac{12}{30} + \frac{10}{30} = \frac{12 + 10}{30} = \frac{22}{30}$.
This fraction cannot be simplified further as 22 and 30 have no common factors besides 1.

Therefore, the sum of $\frac{4}{10} + \frac{2}{6}$ is $\frac{22}{30}$.