Operations with Fractions

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 the problem of adding $\frac{1}{4}$ and $\frac{3}{6}$, we perform the following steps:

• Step 1: Find the least common multiple (LCM) of the denominators $4$ and $6$. The LCM of $4$ and $6$ is $12$.
• Step 2: Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of $12$.
  Multiply both the numerator and denominator of $\frac{1}{4}$ by $3$ to get $\frac{3}{12}$.
• Step 3: Convert $\frac{3}{6}$ to an equivalent fraction with a denominator of $12$.
  Multiply both the numerator and denominator of $\frac{3}{6}$ by $2$ to get $\frac{6}{12}$.
• Step 4: Add the equivalent fractions $\frac{3}{12} + \frac{6}{12}$.
• Step 5: Combine the numerators while keeping the common denominator: $\frac{3+6}{12} = \frac{9}{12}$.
• Step 6: Simplify $\frac{9}{12}$ by dividing the numerator and the denominator by their greatest common divisor, which is $3$, resulting in $\frac{3}{4}$.

Therefore, the sum of $\frac{1}{4}$ and $\frac{3}{6}$ is $\frac{3}{4}$.

To solve the problem of adding the fractions $\frac{1}{5}$ and $\frac{1}{3}$, we follow these steps:

• Step 1: Find a common denominator for the fractions. Since the denominators are $5$ and $3$, the least common multiple is $15$.
• Step 2: Convert each fraction to this common denominator:
  - For $\frac{1}{5}$, multiply both numerator and denominator by $3$ (the denominator of the other fraction), resulting in $\frac{3}{15}$.
  - For $\frac{1}{3}$, multiply both numerator and denominator by $5$ (the denominator of the other fraction), resulting in $\frac{5}{15}$.
• Step 3: Add the fractions now that they have a common denominator:
  $\frac{3}{15} + \frac{5}{15} = \frac{3+5}{15} = \frac{8}{15}$.

Therefore, when you add $\frac{1}{5}$ and $\frac{1}{3}$, the solution is $\frac{8}{15}$.

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 $\frac{13}{20}$.

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 $\frac{11}{12}$.