Division of Fractions - Examples, Exercises and Solutions

When we approach solving such questions, we need to know the rule of dividing fractions,

When we need to divide a fraction by a fraction, we use the method of multiplying by the reciprocal.

This means we flip the numerator and denominator of the second fraction, and then perform fraction multiplication.

Instead of:

1/4 : 1/2 =

We get:

1/4 * 2/1 =

We'll remember that in fraction multiplication we multiply numerator by numerator and denominator by denominator

1*2 / 4*1 =
2/4 =

We'll reduce the fraction and get:

1/2

To solve the division of the fractions $\frac{1}{9}$ and $\frac{1}{3}$, we'll employ the method of "invert and multiply":

• Step 1: Identify the reciprocal of the divisor. The divisor is $\frac{1}{3}$, and its reciprocal is $\frac{3}{1}$.
• Step 2: Convert the division into a multiplication. Therefore, $\frac{1}{9} \div \frac{1}{3}$ becomes $\frac{1}{9} \times \frac{3}{1}$.
• Step 3: Carry out the multiplication of the two fractions.
  $\frac{1}{9} \times \frac{3}{1} = \frac{1 \times 3}{9 \times 1} = \frac{3}{9}$.
• Step 4: Simplify the resulting fraction.
  $\frac{3}{9}$ simplifies to $\frac{1}{3}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

Therefore, the solution to the problem $\frac{1}{9} \div \frac{1}{3}$ is $\frac{1}{3}$.

To solve the fraction division $\frac{8}{9} \div \frac{2}{3}$, follow these steps:

• Step 1: Identify the given fractions $\frac{8}{9}$ and $\frac{2}{3}$.
• Step 2: Find the reciprocal of the second fraction. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
• Step 3: Multiply the first fraction by the reciprocal of the second fraction: $\frac{8}{9} \times \frac{3}{2}$.
• Step 4: Perform the multiplication: multiply the numerators together and the denominators together.

Let's compute the multiplication:

$\frac{8}{9} \times \frac{3}{2} = \frac{8 \times 3}{9 \times 2} = \frac{24}{18}$

Step 5: Simplify the resulting fraction $\frac{24}{18}$.

To simplify, find the greatest common divisor (GCD) of 24 and 18, which is 6. Divide both the numerator and the denominator by 6:

$\frac{24}{18} = \frac{24 \div 6}{18 \div 6} = \frac{4}{3}$

Step 6: If necessary, convert the improper fraction to a mixed number.

Since $\frac{4}{3}$ is an improper fraction, it can be converted to a mixed number:

$\frac{4}{3} = 1 \frac{1}{3}$

Therefore, the solution to the problem $\frac{8}{9} : \frac{2}{3}$ is $1 \frac{1}{3}$.

To solve this problem, we'll break it into these manageable steps:

• Step 1: Identify the fractions:
  $\frac{3}{4}$ and $\frac{1}{2}$.
• Step 2: Find the reciprocal of the second fraction:
• The reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$.
• Step 3: Change the division into multiplication:
• $\frac{3}{4} \div \frac{1}{2}$ becomes $\frac{3}{4} \times \frac{2}{1}$.
• Step 4: Multiply the numerators and the denominators:
• Numerator: $3 \times 2 = 6$
• Denominator: $4 \times 1 = 4$
• So, $\frac{3}{4} \times \frac{2}{1} = \frac{6}{4}$.
• Step 5: Simplify the fraction:
• $\frac{6}{4} = \frac{3}{2}$, since dividing numerator and denominator by 2 gives $\frac{3}{2}$.
• Step 6: Convert the fraction to a mixed number:
• $\frac{3}{2}$ can be written as the mixed number $1\frac{1}{2}$.

Therefore, the result of the division is $1\frac{1}{2}$.

To solve the division of fractions problem $\frac{1}{6} \div \frac{1}{3}$, we'll apply the concept of multiplying by the reciprocal.

• Step 1: Identify the reciprocal of the second fraction. The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.
• Step 2: Multiply the first fraction by this reciprocal. Therefore, calculate $\frac{1}{6} \times \frac{3}{1}$.
• Step 3: Perform the multiplication. Multiply the numerators: $1 \times 3 = 3$. Multiply the denominators: $6 \times 1 = 6$.
• Step 4: Simplify the resulting fraction. The fraction $\frac{3}{6}$ simplifies to $\frac{1}{2}$ because both the numerator and denominator can be divided by 3.

Therefore, the solution to the problem is $\frac{1}{2}$.

To solve the division of fractions $\frac{2}{4} : \frac{2}{2}$, follow these steps:

• Step 1: Identify the fractions — the first fraction is $\frac{2}{4}$, and the second fraction is $\frac{2}{2}$.
• Step 2: Find the reciprocal of the second fraction. The reciprocal of $\frac{2}{2}$ is $\frac{2}{2}$, as it simplifies to 1.
• Step 3: Multiply the first fraction by the reciprocal of the second fraction:

$\frac{2}{4} \times \frac{2}{2} = \frac{2 \times 2}{4 \times 2} = \frac{4}{8}$

Step 4: Simplify the resulting fraction $\frac{4}{8}$. Since the greatest common divisor of 4 and 8 is 4, divide both numerator and denominator by 4:

$\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}$

Therefore, the solution to the problem is $\frac{1}{2}$.

To solve the division problem $\frac{4}{12}:\frac{2}{4}$, we will follow these steps:

• Step 1: Identify the fractions involved: $\frac{4}{12}$ and $\frac{2}{4}$.
• Step 2: Convert the division into multiplication by the reciprocal of the divisor. The reciprocal of $\frac{2}{4}$ is $\frac{4}{2}$.
• Step 3: Multiply the first fraction by this reciprocal:

$\frac{4}{12} \times \frac{4}{2} = \frac{4 \times 4}{12 \times 2}$

$= \frac{16}{24}$

• Step 4: Simplify the resulting fraction. The greatest common divisor of 16 and 24 is 8.

$\frac{16 \div 8}{24 \div 8} = \frac{2}{3}$

Thus, the solution to the problem is $\frac{2}{3}$.

To solve the division of two fractions $\frac{1}{2} \div \frac{1}{2}$, we follow these steps:

• Step 1: Recognize that dividing by a fraction is equivalent to multiplying by its reciprocal. In this case, we replace division with multiplication by flipping the second fraction.
• Step 2: Thus, $\frac{1}{2} \div \frac{1}{2}$ becomes $\frac{1}{2} \times \frac{2}{1}$.
• Step 3: Perform the multiplication: Multiply the numerators and the denominators.
  Numerator: $1 \times 2 = 2$
  Denominator: $2 \times 1 = 2$
• Step 4: Simplify the result: The fraction $\frac{2}{2}$ simplifies to 1.

Thus, the result of the division $\frac{1}{2} \div \frac{1}{2}$ is $1$.

According to the rules of the order of operations, we should first solve the exercise from left to right since there are only multiplication and division operations present:

$1\times\frac{1}{2}=\frac{1}{2}$

$\frac{1}{2}:2=\frac{1}{4}$

To solve the division $\frac{6}{10} \div \frac{2}{5}$, we use the method of multiplying by the reciprocal of the divisor.

First, identify the reciprocal of the divisor $\frac{2}{5}$. The reciprocal is obtained by swapping the numerator and denominator, resulting in $\frac{5}{2}$.

Next, replace the division operation with multiplication by the reciprocal:

$\frac{6}{10} \times \frac{5}{2}$.

Now, perform the multiplication of the fractions by multiplying numerator by numerator and denominator by denominator:

$\frac{6 \times 5}{10 \times 2} = \frac{30}{20}$.

Simplify the fraction $\frac{30}{20}$ by finding the greatest common divisor. Both 30 and 20 can be divided by 10:

$\frac{30 \div 10}{20 \div 10} = \frac{3}{2}$.

The simplified fraction $\frac{3}{2}$ can also be expressed as a mixed number:

$1\frac{1}{2}$.

Therefore, the solution to the problem $\frac{6}{10} \div \frac{2}{5}$ is $1\frac{1}{2}$, aligning with choice 1.

To solve the division of fractions problem $\frac{3}{4}:\frac{1}{6}$, we will use multiplication by the reciprocal. Let us break down the steps:

• Step 1: Identify the fractions. The first fraction is $\frac{3}{4}$ and the second fraction is $\frac{1}{6}$.
• Step 2: Find the reciprocal of the second fraction. The reciprocal of $\frac{1}{6}$ is $\frac{6}{1}$ or just $6$.
• Step 3: Multiply the first fraction by the reciprocal of the second fraction:

$\frac{3}{4} \times \frac{6}{1} = \frac{3 \times 6}{4 \times 1} = \frac{18}{4}$

• Step 4: Simplify the resulting fraction. Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2:

$\frac{18}{4} = \frac{18 \div 2}{4 \div 2} = \frac{9}{2}$

• Step 5: Convert the improper fraction to a mixed number. Divide 9 by 2, which gives 4 with a remainder of 1:

$\frac{9}{2} = 4\frac{1}{2}$

Thus, the solution to the problem $\frac{3}{4}:\frac{1}{6}$ is $4\frac{1}{2}$.

To solve this problem, we'll employ the division of fractions technique:

• Step 1: Find the reciprocal of the divisor. Here, the divisor is $\frac{4}{7}$. Its reciprocal is $\frac{7}{4}$.
• Step 2: Multiply the dividend $\frac{2}{5}$ by the reciprocal of the divisor.
• Step 3: Calculate $\frac{2}{5} \times \frac{7}{4}$.
• Step 4: Simplify the result, if possible.

Now, let's work through these steps:

Step 1: The reciprocal of $\frac{4}{7}$ is $\frac{7}{4}$.

Step 2: Multiply the fractions: $\frac{2}{5} \times \frac{7}{4} = \frac{2 \times 7}{5 \times 4} = \frac{14}{20}$.

Step 3: Simplify $\frac{14}{20}$. The greatest common divisor (GCD) of 14 and 20 is 2, so divide both the numerator and the denominator by 2:

$\frac{14}{20} = \frac{14 \div 2}{20 \div 2} = \frac{7}{10}$.

Therefore, the solution to the problem is $\frac{7}{10}$.

To solve this problem, let's follow these steps:

• Step 1: Simplify the fraction $\frac{2}{4}$.
• Step 2: Find the reciprocal of $\frac{1}{3}$.
• Step 3: Multiply the simplified fraction by the reciprocal.
• Step 4: Simplify the resulting fraction if needed.

Now, let's work through each step:
Step 1: Simplify $\frac{2}{4}$ to $\frac{1}{2}$ by dividing both the numerator and the denominator by 2.
Step 2: The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.
Step 3: Multiply $\frac{1}{2} \times \frac{3}{1}$. This gives us $\frac{1 \times 3}{2 \times 1} = \frac{3}{2}$.
Step 4: $\frac{3}{2}$ is an improper fraction. Convert it to a mixed number, $1\frac{1}{2}$.

Therefore, the solution to the division $\frac{2}{4}:\frac{1}{3}$ is $1\frac{1}{2}$.

To solve the division $\frac{1}{2} \div \frac{3}{5}$, we will follow the multiplication by the reciprocal method. Here are the steps:

• Step 1: Find the reciprocal of the divisor $\frac{3}{5}$, which is $\frac{5}{3}$.
• Step 2: Multiply the dividend $\frac{1}{2}$ by the reciprocal found in Step 1: $\frac{1}{2} \times \frac{5}{3}$.
• Step 3: Multiply the numerators: $1 \times 5 = 5$.
• Step 4: Multiply the denominators: $2 \times 3 = 6$.
• Step 5: Combine the results to form the fraction $\frac{5}{6}$.

The simplified result of $\frac{1}{2} \div \frac{3}{5}$ is $\frac{5}{6}$.

To solve this problem, we need to divide the fraction $\frac{1}{5}$ by the fraction $\frac{1}{10}$. When dividing fractions, the procedure involves multiplying by the reciprocal of the divisor (the second fraction).

Let's start with the solution:

• First, determine the reciprocal of $\frac{1}{10}$. The reciprocal of a fraction is obtained by swapping its numerator and denominator. Thus, the reciprocal of $\frac{1}{10}$ is $\frac{10}{1}$.
• Now multiply $\frac{1}{5}$ by the reciprocal of $\frac{1}{10}$, which is $\frac{10}{1}$:

$\frac{1}{5} \times \frac{10}{1} = \frac{1 \times 10}{5 \times 1} = \frac{10}{5}$

Simplify the fraction $\frac{10}{5}$:

$\frac{10}{5} = 2$

Therefore, the result of $\frac{1}{5} : \frac{1}{10}$ is $2$.