Addition of Fractions: Finding a Common Denominator by Multiplying the Denominators

Let's try to find the least common denominator between 10 and 3

To find the least common denominator, we need to find a number that is divisible by both 10 and 3

In this case, the common denominator is 30

Now we'll multiply each fraction by the appropriate number to reach the denominator 30

We'll multiply the first fraction by 3

We'll multiply the second fraction by 10

$\frac{2\times3}{10\times3}+\frac{1\times10}{3\times10}=\frac{6}{30}+\frac{10}{30}$

Now we'll combine and get:

$\frac{6+10}{30}=\frac{16}{30}$

Let's try to find the least common denominator between 5 and 3

To find the least common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

$\frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}$

Now we'll combine and get:

$\frac{6+5}{15}=\frac{11}{15}$

Let's try to find the least common multiple (LCM) between 8 and 3

To find the least common multiple, we need to find a number that is divisible by both 8 and 3

In this case, the common multiple is 24

Now we'll 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 8

$\frac{2\times3}{8\times3}+\frac{2\times8}{3\times8}=\frac{6}{24}+\frac{16}{24}$

Now we'll combine and get:

$\frac{6+16}{24}=\frac{22}{24}$

Let's try to find the least common multiple (LCM) between 8 and 3

To find the least common multiple, we need to find a number that is divisible by both 8 and 3

In this case, the common multiple is 24

Now we'll 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 8

$\frac{3\times3}{8\times3}+\frac{2\times8}{3\times8}=\frac{9}{24}+\frac{16}{24}$

Now we'll combine and get:

$\frac{9+16}{24}=\frac{25}{24}$

Let's try to find the lowest common denominator between 2 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 2 and 9

In this case, the common denominator is 18

Now we'll multiply each fraction by the appropriate number to reach the denominator 18

We'll multiply the first fraction by 9

We'll multiply the second fraction by 2

$\frac{1\times9}{2\times9}+\frac{2\times2}{9\times2}=\frac{9}{18}+\frac{4}{18}$

Now we'll combine and get:

$\frac{9+4}{18}=\frac{13}{18}$

Let's try to find the lowest common denominator between 4 and 6

To find 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

Now we'll 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

$\frac{1\times3}{4\times3}+\frac{2\times2}{6\times2}=\frac{3}{12}+\frac{4}{12}$

Now we'll combine and get:

$\frac{3+4}{12}=\frac{7}{12}$

Let's try to find the lowest common denominator between 4 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 9

In this case, the common denominator is 36

Now we'll multiply each fraction by the appropriate number to reach the denominator 36

We'll multiply the first fraction by 9

We'll multiply the second fraction by 4

$\frac{1\times9}{4\times9}+\frac{1\times4}{9\times4}=\frac{9}{36}+\frac{4}{36}$

Now we'll combine and get:

$\frac{9+4}{36}=\frac{13}{36}$

Let's try to find the lowest common denominator between 5 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

$\frac{1\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{3}{15}+\frac{5}{15}$

Now we'll combine and get:

$\frac{3+5}{15}=\frac{8}{15}$

Let's try to find the lowest common denominator between 5 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

$\frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}$

Now we'll combine and get:

$\frac{6+5}{15}=\frac{11}{15}$

Let's try to find the lowest common denominator between 5 and 4

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 4

In this case, the common denominator is 20

Now we'll multiply each fraction by the appropriate number to reach the denominator 20

We'll multiply the first fraction by 4

We'll multiply the second fraction by 5

$\frac{2\times4}{5\times4}+\frac{1\times5}{4\times5}=\frac{8}{20}+\frac{5}{20}$

Now we'll combine and get:

$\frac{8+5}{20}=\frac{13}{20}$

Let's try to find the lowest common denominator between 5 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 6

In this case, the common denominator is 30

Now we'll multiply each fraction by the appropriate number to reach the denominator 30

We'll multiply the first fraction by 6

We'll multiply the second fraction by 5

$\frac{2\times6}{5\times6}+\frac{1\times5}{6\times5}=\frac{12}{30}+\frac{5}{30}$

Now we'll combine and get:

$\frac{12+5}{30}=\frac{17}{30}$

Let's try to find the lowest common denominator between 5 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 6

In this case, the common denominator is 30

Now we'll multiply each fraction by the appropriate number to reach the denominator 30

We'll multiply the first fraction by 6

We'll multiply the second fraction by 5

$\frac{2\times6}{5\times6}+\frac{3\times5}{6\times5}=\frac{12}{30}+\frac{15}{30}$

Now we'll combine and get:

$\frac{12+15}{30}=\frac{27}{30}$

Let's try to find the lowest common denominator between 7 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 7 and 3

In this case, the common denominator is 21

Now we'll multiply each fraction by the appropriate number to reach the denominator 21

We'll multiply the first fraction by 3

We'll multiply the second fraction by 7

$\frac{1\times3}{7\times3}+\frac{1\times7}{3\times7}=\frac{3}{21}+\frac{7}{21}$

Now we'll combine and get:

$\frac{3+7}{21}=\frac{10}{21}$

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 addition of the fractions $\frac{2}{9}$ and $\frac{1}{2}$, follow these steps:

• Step 1: Determine the Common Denominator.
  The least common denominator for 9 and 2 is $18$ because $9 \times 2 = 18$.
• Step 2: Adjust Each Fraction.
  Convert $\frac{2}{9}$ to a fraction over 18. Multiply both the numerator and the denominator by 2:
  $\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}$.
  Convert $\frac{1}{2}$ to a fraction over 18. Multiply both the numerator and the denominator by 9:
  $\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}$.
• Step 3: Add the Fractions.
  Add the resulting fractions: $\frac{4}{18} + \frac{9}{18} = \frac{4 + 9}{18} = \frac{13}{18}$.

Thus, the sum of $\frac{2}{9}$ and $\frac{1}{2}$ is $\frac{13}{18}$.

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 given problem, we will follow these steps:

• Step 1: Determine a common denominator for the fractions.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Add the numerators and keep the common denominator.
• Step 4: Simplify the resulting fraction if possible.

Let's proceed with each step:

Step 1: Determine a common denominator.
The denominators of the fractions are 5 and 7. The least common multiple (LCM) of 5 and 7 is 35. Thus, the common denominator is 35.

Step 2: Convert each fraction to have the common denominator of 35.
Convert $\frac{3}{5}$ to a fraction with a denominator of 35: $\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$.
Convert $\frac{2}{7}$ to a fraction with a denominator of 35: $\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}$.

Step 3: Add the numerators and use the common denominator.
Now add the fractions: $\frac{21}{35} + \frac{10}{35} = \frac{21+10}{35} = \frac{31}{35}$.

Step 4: Simplify the result.
The fraction $\frac{31}{35}$ is already in its simplest form since 31 and 35 have no common factors other than 1.

Therefore, the solution to the problem is $\frac{31}{35}$.

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 $\frac{13}{20}$.