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

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:

$\frac{5}{15} - \frac{3}{15} = \frac{5 - 3}{15} = \frac{2}{15}$

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

To solve the subtraction problem $\frac{3}{7} - \frac{1}{6}$, we need to follow these steps:

• Step 1: Identify the least common multiple (LCM) of the denominators. For 7 and 6, the LCM is 42.
• Step 2: Convert each fraction to an equivalent fraction with a denominator of 42.
• For $\frac{3}{7}$, multiply the numerator and the denominator by 6 to get $\frac{18}{42}$.
• For $\frac{1}{6}$, multiply the numerator and the denominator by 7 to get $\frac{7}{42}$.
• Step 3: Subtract the two fractions: $\frac{18}{42} - \frac{7}{42} = \frac{11}{42}$.
• Step 4: Check if the fraction $\frac{11}{42}$ can be simplified. Since 11 and 42 have no common factors besides 1, the fraction is already in its simplest form.

Therefore, the solution to the problem is $\frac{11}{42}$.

Among the provided choices, the correct answer is: $\frac{11}{42}$.

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 given problem $\frac{1}{2} - \frac{2}{7}$, we need to follow these steps:

• Step 1: Find a common denominator for the fractions. The denominators are 2 and 7. The common denominator can be found by multiplying these two numbers: $2 \times 7 = 14$.
• Step 2: Convert the fractions to equivalent fractions with the common denominator. For $\frac{1}{2}$, multiply the numerator and the denominator by 7: $\frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}$. For $\frac{2}{7}$, multiply the numerator and the denominator by 2: $\frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14}$.
• Step 3: Subtract the fractions. With a common denominator, subtract the numerators: $\frac{7}{14} - \frac{4}{14} = \frac{7 - 4}{14} = \frac{3}{14}$.
• Step 4: Simplify the resulting fraction if needed. The fraction $\frac{3}{14}$ is already in its simplest form.

Thus, the solution to the problem is $\frac{3}{14}$.

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 the subtraction of these two fractions, we follow these 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 each fraction to have this common denominator:
  • For $\frac{3}{4}$, multiply both the numerator and denominator by 3 to get $\frac{9}{12}$.
  • For $\frac{3}{6}$, multiply both the numerator and denominator by 2 to get $\frac{6}{12}$.
• Step 3: Perform the subtraction: $\frac{9}{12} - \frac{6}{12} = \frac{3}{12}$.
• Step 4: Simplify the fraction $\frac{3}{12}$. The greatest common divisor (GCD) of 3 and 12 is 3. Divide both numerator and denominator by 3 to get $\frac{1}{4}$.

Therefore, the solution to the problem is $\frac{1}{4}$.

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 subtracting $\frac{1}{3}$ from $\frac{3}{8}$, follow these steps:

Step 1: Identify the denominators of the fractions, which are 8 and 3, respectively. The least common denominator (LCD) is the product of these two denominators, as they have no common factors. Thus, the LCD is $8 \times 3 = 24$.

Step 2: Convert each fraction to an equivalent form with the common denominator 24.

• Convert $\frac{3}{8}$ to an equivalent fraction with a denominator of 24 by multiplying both the numerator and the denominator by $\frac{24}{8} = 3$:
  $\frac{3}{8} \times \frac{3}{3} = \frac{9}{24}$.
• Convert $\frac{1}{3}$ to an equivalent fraction with a denominator of 24 by multiplying both the numerator and the denominator by $\frac{24}{3} = 8$:
  $\frac{1}{3} \times \frac{8}{8} = \frac{8}{24}$.

Step 3: Subtract the numerators of these equivalent fractions, maintaining the common denominator:

$\frac{9}{24} - \frac{8}{24} = \frac{9 - 8}{24} = \frac{1}{24}$.

Therefore, the solution to $\frac{3}{8} - \frac{1}{3}$ is $\frac{1}{24}$.

To solve this problem, we'll walk through these steps:

• Step 1: Determine the least common denominator (LCD) for the denominators 5 and 6.
• Step 2: Convert each fraction to an equivalent fraction with the LCD.
• Step 3: Subtract the numerators of the equivalent fractions.
• Step 4: Simplify the resulting fraction if possible.

Now, let's work through each step:

Step 1: Determine the least common denominator (LCD).
The denominators are 5 and 6. Since there is no common factor, the LCD is $5 \times 6 = 30$.

Step 2: Convert each fraction to an equivalent fraction with the LCD.
For $\frac{2}{5}$, multiply numerator and denominator by 6 to get $\frac{2 \times 6}{5 \times 6} = \frac{12}{30}$.
For $\frac{2}{6}$, multiply numerator and denominator by 5 to get $\frac{2 \times 5}{6 \times 5} = \frac{10}{30}$.

Step 3: Subtract the fractions with the same denominator.
$\frac{12}{30} - \frac{10}{30} = \frac{12 - 10}{30} = \frac{2}{30}$.

Step 4: Simplify the resulting fraction.
$\frac{2}{30}$ can be simplified by dividing numerator and denominator by their greatest common divisor, which is 2.
Thus, $\frac{2}{30} = \frac{1}{15}$.

Therefore, the solution to the problem is $\frac{1}{15}$.

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

• Step 1: Identify the fractions and denominators involved.
• Step 2: Find a common denominator for the two fractions.
• Step 3: Convert each fraction to an equivalent fraction with the common denominator.
• Step 4: Subtract the numerators and simplify the result.

Let's perform each of these steps:

Step 1: We have the fractions $\frac{3}{7}$ and $\frac{1}{3}$.

Step 2: Find a common denominator. The denominators are 7 and 3, so the common denominator will be $7 \times 3 = 21$.

Step 3: Convert each fraction:

• For $\frac{3}{7}$, multiply both numerator and denominator by 3 to get $\frac{3 \times 3}{7 \times 3} = \frac{9}{21}$.
• For $\frac{1}{3}$, multiply both numerator and denominator by 7 to get $\frac{1 \times 7}{3 \times 7} = \frac{7}{21}$.

Step 4: Subtract the numerators:

$\frac{9}{21} - \frac{7}{21} = \frac{9 - 7}{21} = \frac{2}{21}$.

Simplify if necessary: Here, $\frac{2}{21}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{2}{21}$.

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

• Step 1: Find a common denominator for the fractions $\frac{3}{4}$ and $\frac{3}{9}$.
• Step 2: Convert each fraction to have this common denominator.
• Step 3: Subtract the numerators while keeping the common denominator.
• Step 4: Simplify the resultant fraction if possible.

Now, let's work through each step:

Step 1: The denominators of the fractions are 4 and 9. The least common multiple of 4 and 9 is 36. Therefore, 36 will be our common denominator.

Step 2: Convert $\frac{3}{4}$ and $\frac{3}{9}$ to have a denominator of 36:

$\frac{3}{4} \times \frac{9}{9} = \frac{27}{36}$ and $\frac{3}{9} \times \frac{4}{4} = \frac{12}{36}$

Step 3: Now, subtract the fractions:

$\frac{27}{36} - \frac{12}{36} = \frac{15}{36}$

Step 4: Simplify $\frac{15}{36}$:

Both 15 and 36 can be divided by their greatest common divisor, which is 3. Dividing both the numerator and denominator by 3, we get:

$\frac{15 \div 3}{36 \div 3} = \frac{5}{12}$

Therefore, the solution to the problem is $\frac{5}{12}$.

To solve the subtraction of these two fractions, we'll follow these steps:

• Step 1: Find a common denominator for the fractions $\frac{4}{5}$ and $\frac{1}{3}$. The denominators are 5 and 3, and multiplying them gives us a common denominator of 15.
• Step 2: Convert each fraction to have this common denominator:
• Convert $\frac{4}{5}$ to an equivalent fraction with a denominator of 15. To do this, multiply both the numerator and the denominator by 3: $\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}$.
• Convert $\frac{1}{3}$ to an equivalent fraction with a denominator of 15. To do this, multiply both the numerator and the denominator by 5: $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$.
• Step 3: Perform the subtraction of the numerators, now having the same denominator: $\frac{12}{15} - \frac{5}{15} = \frac{12 - 5}{15} = \frac{7}{15}$.
• Step 4: Simplify the result, if possible. In this case, $\frac{7}{15}$ is already in its simplest form as 7 and 15 have no common factors other than 1.

Therefore, the solution to the problem is $\frac{7}{15}$.

To solve $\frac{6}{10} - \frac{1}{3}$, we need to perform the following steps:

• Step 1: Identify a common denominator. The denominators 10 and 3 have a least common multiple of 30. We will use 30 as the common denominator.
• Step 2: Convert the fractions to have the common denominator of 30.
  $\frac{6}{10}$ needs to be converted by multiplying both the numerator and the denominator by 3: $\frac{6 \times 3}{10 \times 3} = \frac{18}{30}$ $\frac{1}{3}$ needs to be converted by multiplying both the numerator and the denominator by 10: $\frac{1 \times 10}{3 \times 10} = \frac{10}{30}$
• Step 3: Subtract the new fractions: $\frac{18}{30} - \frac{10}{30} = \frac{18 - 10}{30} = \frac{8}{30}$
• Step 4: Simplify the fraction $\frac{8}{30}$.
  Divide both the numerator and the denominator by their greatest common divisor, which is 2: $\frac{8 \div 2}{30 \div 2} = \frac{4}{15}$

Therefore, the result of the subtraction is $\frac{4}{15}$.