Operations with Fractions - Examples, Exercises and Solutions

To solve the problem $\frac{1}{3} - \frac{1}{5}$, we follow these steps:

First, we need to find a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{5}$. The denominators are 3 and 5, and their least common multiple (LCM) is 15.

We will convert each fraction to an equivalent fraction with the denominator 15:

• To convert $\frac{1}{3}$ to a fraction with denominator 15, multiply both the numerator and the denominator by 5: $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$
• To convert $\frac{1}{5}$ to a fraction with denominator 15, multiply both the numerator and the denominator by 3: $\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

Now that both fractions have the same denominator, we can subtract the numerators:

Therefore, the solution to the problem is $\frac{2}{15}$.

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

• Step 1: Find the common denominator for the fractions $\frac{2}{4}$ and $\frac{1}{3}$.
• Step 2: Convert each fraction to have the common denominator.
• Step 3: Perform the subtraction and simplify if necessary.

Now, let's work through these steps:

Step 1: The denominators are $4$ and $3$. The common denominator is the product $4 \times 3 = 12$.

Step 2: Convert each fraction:
$\frac{2}{4} = \frac{2 \times 3}{4 \times 3} = \frac{6}{12}$
$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$

Step 3: Subtract the fractions with a common denominator:
$\frac{6}{12} - \frac{4}{12} = \frac{6 - 4}{12} = \frac{2}{12}$

Finally, simplify $\frac{2}{12}$. The greatest common divisor of 2 and 12 is 2, so:
$\frac{2}{12} = \frac{2 \div 2}{12 \div 2} = \frac{1}{6}$

Therefore, the solution to the problem is $\frac{1}{6}$.

To solve the subtraction of fractions $\frac{3}{5} - \frac{1}{2}$, we will follow these steps:

• Step 1: Find the least common multiple (LCM) of the denominators 5 and 2. The LCM of 5 and 2 is 10.
• Step 2: Convert each fraction to have a denominator of 10.
• Step 3: Subtract the converted fractions.
• Step 4: Simplify the result if necessary.

Now, let's work through each step in detail:

Step 1: The LCM of 5 and 2 is 10, since 10 is the smallest number that both 5 and 2 divide into evenly.

Step 2: Convert each fraction to have a denominator of 10.

For $\frac{3}{5}$:
Multiply numerator and denominator by 2 to get $\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$.

For $\frac{1}{2}$:
Multiply numerator and denominator by 5 to get $\frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

Step 3: Subtract the fractions:

$\frac{6}{10} - \frac{5}{10} = \frac{6 - 5}{10} = \frac{1}{10}$.

Step 4: There is no further simplification needed for $\frac{1}{10}$ as it is already in its simplest form.

Therefore, the solution to the problem is $\frac{1}{10}$.

The correct answer, choice (4), is $\frac{1}{10}$.

To solve the problem of subtracting $\frac{1}{4}$ from $\frac{3}{5}$, we need a common denominator.

First, find the least common denominator (LCD) of 5 and 4, which is 20. This is done by multiplying the denominators: $5 \times 4 = 20$.

Next, convert each fraction to an equivalent fraction with the denominator of 20:

• For $\frac{3}{5}$: Multiply both numerator and denominator by 4 to get $\frac{3 \times 4}{5 \times 4} = \frac{12}{20}$.
• For $\frac{1}{4}$: Multiply both numerator and denominator by 5 to get $\frac{1 \times 5}{4 \times 5} = \frac{5}{20}$.

Now perform the subtraction with these equivalent fractions:

$\frac{12}{20} - \frac{5}{20} = \frac{12 - 5}{20} = \frac{7}{20}$

The resulting fraction, $\frac{7}{20}$, is already in its simplest form.

Therefore, the solution to the subtraction $\frac{3}{5} - \frac{1}{4}$ is $\frac{7}{20}$.

Checking against the multiple-choice answers, the correct choice is the first one: $\frac{7}{20}$.

To solve the subtraction of fractions $\frac{3}{5} - \frac{1}{3}$, follow these steps:

• Step 1: Find the Least Common Denominator (LCD)
  The denominators are 5 and 3. The least common multiple of 5 and 3 is 15. Thus, the common denominator will be 15.
• Step 2: Convert fractions to have the same denominator
  For $\frac{3}{5}$, multiply both the numerator and the denominator by 3 to get:
  $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$.
  For $\frac{1}{3}$, multiply both the numerator and the denominator by 5 to get:
  $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Subtract the numerators
  Now subtract the equivalent fractions:
  $\frac{9}{15} - \frac{5}{15} = \frac{9 - 5}{15} = \frac{4}{15}$.
• Step 4: Simplify the fraction
  The fraction $\frac{4}{15}$ is already in its simplest form.

Thus, the solution to the problem is $\frac{4}{15}$.

To solve $\frac{1}{2} - \frac{1}{9}$, follow these steps:

Step 1: Find the least common multiple (LCM) of the denominators 2 and 9.
The multiples of 2 are $2, 4, 6, 8, 10, 12, 14, 16, 18, \ldots$
The multiples of 9 are $9, 18, 27, \ldots$
The smallest common multiple is 18. Thus, the LCM of 2 and 9 is 18.

Step 2: Convert each fraction to an equivalent fraction with the common denominator 18.
For $\frac{1}{2}$, the equivalent fraction with 18 as the denominator is calculated by finding the factor needed:
$\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}$.
For $\frac{1}{9}$, the equivalent fraction with 18 as the denominator is:
$\frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18}$.

Step 3: Perform the subtraction of these equivalent fractions.
$\frac{9}{18} - \frac{2}{18} = \frac{9 - 2}{18} = \frac{7}{18}$.

Therefore, the solution to the problem is $\boxed{\frac{7}{18}}$.

To solve the problem of adding $\frac{4}{9}$ and $\frac{1}{2}$, we'll proceed step-by-step:

• Step 1: Determine a common denominator.
• Step 2: Convert each fraction to an equivalent fraction with this common denominator.
• Step 3: Add the numerators of these converted fractions.
• Step 4: Simplify the resulting fraction if necessary.

Now, let's perform these steps in detail:

Step 1: Determine the common denominator.
The denominators are 9 and 2. The least common denominator (LCD) can be found by multiplying these because they have no common factors other than 1:
$\text{LCD} = 9 \times 2 = 18$.

Step 2: Convert each fraction to have the common denominator of 18.

• Convert $\frac{4}{9}$: $\frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18}$
• Convert $\frac{1}{2}$: $\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18}$

Step 3: Add the numerators of the converted fractions:
$\frac{8}{18} + \frac{9}{18} = \frac{8+9}{18} = \frac{17}{18}$

Step 4: Simplification (if needed):
The fraction $\frac{17}{18}$ is already in its simplest form.

Therefore, the sum of $\frac{4}{9}$ and $\frac{1}{2}$ is $\frac{17}{18}$.

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 $\frac{1}{2}$.

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 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 this problem, we'll begin by finding a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{4}$.
Step 1: Identify the denominators, which are 3 and 4. Multiply these to get a common denominator: $3 \times 4 = 12$.

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 12.

• To convert $\frac{1}{3}$ to a denominator of 12, multiply both the numerator and the denominator by 4: $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$.
• To convert $\frac{1}{4}$ to a denominator of 12, multiply both the numerator and the denominator by 3: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.

Step 3: Add the resulting fractions: $\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}$.

Thus, the sum of $\frac{1}{3}$ and $\frac{1}{4}$ is $\frac{7}{12}$.

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 $\frac{2}{3}$.