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

Let's first identify the lowest common denominator between 10 and 12.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 10 and 12.

In this case, the common denominator is 60.

We'll proceed to multiply each fraction by the appropriate number to reach the denominator 60.

We'll multiply the first fraction by 6

We'll multiply the second fraction by 5

$\frac{4\times6}{10\times6}+\frac{5\times5}{12\times5}=\frac{24}{60}+\frac{25}{60}$

Now let's add:

$\frac{24+25}{60}=\frac{49}{60}$

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:

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

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:

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

Let's try to find the lowest common multiple between 8 and 10

To find the lowest common multiple, we need to find a number that is divisible by both 8 and 10

In this case, the lowest common multiple is 40

Now, let's multiply each number in the appropriate multiples to reach the number 40

We will multiply the first number by 5

We will multiply the second number by 4

$\frac{4\times5}{8\times5}+\frac{4\times4}{10\times4}=\frac{20}{40}+\frac{16}{40}$

Now let's calculate:

$\frac{20+16}{40}=\frac{36}{40}$

We must first identify the lowest common denominator between 4 and 6.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 4 and 6.

In this case, the common denominator is 12.

We will then proceed to multiply each fraction by the appropriate number to reach the denominator 12

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

$\frac{1\times3}{4\times3}+\frac{3\times2}{6\times2}=\frac{3}{12}+\frac{6}{12}$

Finally we'll combine and obtain the following:

$\frac{6+3}{12}=\frac{9}{12}$

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 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 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 $\frac{22}{30}$.

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 $\frac{12}{18}$.

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 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 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 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 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 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 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 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 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 $\mathbf{\frac{9}{10}}$.