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

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:

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

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 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 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 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 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:

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

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 $\frac{2}{3}$.

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 $\frac{3}{14} + \frac{3}{7}$, 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 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 $\frac{19}{24}$.