Division of Fractions: Division by a whole number

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 this problem, we'll convert the division of the fraction $\frac{9}{13}$ by the whole number 3 into a multiplication problem. This involves multiplying by the reciprocal of 3.

Step-by-step solution:
Step 1: Rewrite the division problem using the concept of reciprocal:
$\frac{9}{13} : 3 = \frac{9}{13} \times \frac{1}{3}$ Step 2: Perform the multiplication of the fractions:
When multiplying fractions, multiply the numerators together and the denominators together:
$\frac{9 \times 1}{13 \times 3} = \frac{9}{39}$ Step 3: Simplify the resulting fraction:
To simplify $\frac{9}{39}$, find the greatest common divisor (GCD) of 9 and 39, which is 3:
Divide both the numerator and the denominator by 3:
$\frac{9 \div 3}{39 \div 3} = \frac{3}{13}$

Therefore, the solution to the problem is $\frac{3}{13}$.

We need to solve the expression $\frac{3}{5} : 3$.

The division of a fraction by a whole number can be rewritten as the multiplication of the fraction by the reciprocal of the whole number.

The given problem transforms as follows:

$\frac{3}{5} : 3 = \frac{3}{5} \times \frac{1}{3}$

Performing the multiplication, we multiply the numerators together and the denominators together:

$\frac{3 \times 1}{5 \times 3} = \frac{3}{15}$

Now, simplify $\frac{3}{15}$ by finding their greatest common divisor (GCD), which is 3:

$\frac{3 \div 3}{15 \div 3} = \frac{1}{5}$

Thus, the answer to the division is $\frac{1}{5}$.

Therefore, the solution to the problem $\frac{3}{5} : 3$ is $\frac{1}{5}$, which corresponds to choice 4.

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

• Step 1: Convert the division into a multiplication problem.
• Step 2: Simplify the result, if necessary.

Now, let's work through each step:
Step 1: We need to divide $\frac{6}{4}$ by 2. We can represent this as:

$\frac{6}{4} \div 2 = \frac{6}{4} \times \frac{1}{2}$

This converts the division into a multiplication problem by using the reciprocal of 2, which is $\frac{1}{2}$.

Step 2: Multiply the fractions:

$\frac{6 \times 1}{4 \times 2} = \frac{6}{8}$

Step 3: Simplify the fraction $\frac{6}{8}$:
To simplify $\frac{6}{8}$, divide both the numerator and the denominator by their greatest common divisor, which is 2:

$\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

Therefore, the solution to the problem is $\frac{3}{4}$.

To solve the problem $\frac{10}{9} : 5$, we will convert the division into a multiplication with a reciprocal.

• Step 1: Recognize that dividing by $5$ is the same as multiplying by $\frac{1}{5}$.
• Step 2: Rewrite the problem as: $\frac{10}{9} \times \frac{1}{5}$.
• Step 3: Multiply the fractions: $\frac{10 \times 1}{9 \times 5} = \frac{10}{45}$.
• Step 4: Simplify the fraction $\frac{10}{45}$ by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 5.
• Step 5: $\frac{10 \div 5}{45 \div 5} = \frac{2}{9}$.

Therefore, the solution to the problem is $\frac{2}{9}$.

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

• Step 1: Simplify the given fraction.
• Step 2: Convert the division by a whole number into multiplication by its reciprocal.
• Step 3: Perform the multiplication and simplify if needed.

Now, let's work through each step:
Step 1: Simplify the fraction $\frac{9}{9}$. Since the numerator and denominator are the same, it simplifies to $1$.
Step 2: Convert the division $1 \div 3$ into multiplication by using the reciprocal: $\frac{1}{3}$. Therefore, $1 \div 3$ becomes $1 \times \frac{1}{3}$.
Step 3: Perform the multiplication: $1 \times \frac{1}{3} = \frac{1}{3}$.

Therefore, the solution to the problem is $\frac{1}{3}$, which corresponds to the answer choice $\frac{1}{3}$.

To solve the equation $\frac{4}{11} \div 4$, let's follow these steps:

• Step 1: Identify the reciprocal of the divisor $4$. The reciprocal of $4$ is $\frac{1}{4}$.
• Step 2: Multiply the original fraction by the reciprocal. Thus, $\frac{4}{11} \times \frac{1}{4}$.
• Step 3: Perform the multiplication. Multiplying two fractions involves multiplying their numerators and their denominators:

$\frac{4}{11} \times \frac{1}{4} = \frac{4 \times 1}{11 \times 4} = \frac{4}{44}$.

Step 4: Simplify the resulting fraction. Notice that $\frac{4}{44}$ can be simplified because 4 and 44 share a common factor of 4:

$\frac{4}{44} = \frac{1}{11}$.

Therefore, the solution to the division problem $\frac{4}{11} \div 4$ is $\frac{1}{11}$.

To solve the problem of dividing the fraction $\frac{8}{9}$ by the whole number 4, we will follow the reciprocal method.

• Step 1: Express the whole number 4 as a fraction: $4 = \frac{4}{1}$.
• Step 2: Find the reciprocal of $\frac{4}{1}$, which is $\frac{1}{4}$.
• Step 3: Multiply the original fraction $\frac{8}{9}$ by $\frac{1}{4}$:
  $\frac{8}{9} \times \frac{1}{4} = \frac{8 \times 1}{9 \times 4} = \frac{8}{36}.$
• Step 4: Simplify the fraction $\frac{8}{36}$. The greatest common divisor (GCD) of 8 and 36 is 4:
  $\frac{8 \div 4}{36 \div 4} = \frac{2}{9}.$

Thus, the result of dividing $\frac{8}{9}$ by 4 is $\frac{2}{9}$.

To solve the problem of dividing $\frac{6}{7}$ by 3, we follow these steps:

• Step 1: Understand that dividing by a number is the same as multiplying by its reciprocal.
• Step 2: Write 3 as a fraction: $3 = \frac{3}{1}$.
• Step 3: Find the reciprocal of $\frac{3}{1}$, which is $\frac{1}{3}$.
• Step 4: Multiply $\frac{6}{7}$ by $\frac{1}{3}$.
• Step 5: Perform the multiplication: $\frac{6}{7} \times \frac{1}{3} = \frac{6 \times 1}{7 \times 3} = \frac{6}{21}$
• Step 6: Simplify $\frac{6}{21}$ by finding the greatest common divisor of 6 and 21, which is 3.
• Step 7: Divide both the numerator and the denominator by 3: $\frac{6 \div 3}{21 \div 3} = \frac{2}{7}$

Thus, the result of the division $\frac{6}{7} \div 3$ is $\frac{2}{7}$.

The correct choice from the provided options is $\frac{2}{7}$, which matches choice number 4.

To solve this problem, we need to divide the fraction $\frac{2}{3}$ by the whole number 2.

We'll follow these steps:

• Step 1: Express the division operation with reciprocal multiplication.
• Step 2: Simplify the expression if possible.

Step 1: To divide $\frac{2}{3}$ by 2, we multiply $\frac{2}{3}$ by the reciprocal of 2. The reciprocal of 2 is $\frac{1}{2}$. Thus, we have:

$\frac{2}{3} \div 2 = \frac{2}{3} \times \frac{1}{2}$

Step 2: Perform the multiplication:

$\frac{2 \times 1}{3 \times 2} = \frac{2}{6}$

Simplify $\frac{2}{6}$ by dividing the numerator and the denominator by their greatest common divisor, which is 2:

$\frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}$

Thus, the solution to the problem is $\frac{1}{3}$.

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

• Step 1: Convert the division of the fraction by the whole number into multiplication by the reciprocal of that whole number.
• Step 2: Calculate the new fraction.
• Step 3: Simplify the fraction if necessary.

Step 1: We have the fraction $\frac{3}{9}$ and we want to divide it by 3. We will convert this division into multiplication by the reciprocal of 3, which is $\frac{1}{3}$. Thus, the operation becomes:

$\frac{3}{9} \times \frac{1}{3}$

Step 2: Perform the multiplication. Multiply the numerators together and the denominators together:

Numerator: $3 \times 1 = 3$
Denominator: $9 \times 3 = 27$

This results in the fraction $\frac{3}{27}$.

Step 3: Simplify $\frac{3}{27}$. Notice that the numerator and the denominator have a common factor of 3:

Divide both the numerator and the denominator by 3:

$\frac{3 \div 3}{27 \div 3} = \frac{1}{9}$

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

To solve the problem of dividing the fraction $\frac{10}{8}$ by the whole number 2, we will follow these steps:

• Step 1: Rewrite the division as multiplication by the reciprocal.
• Step 2: Simplify the resulting fraction, if possible.

Now, let's work through each step:

Step 1: We need to divide $\frac{10}{8}$ by 2. According to the division rule for fractions, dividing by a whole number is equivalent to multiplying by its reciprocal. Therefore, we will multiply $\frac{10}{8}$ by $\frac{1}{2}$:

$\frac{10}{8} \times \frac{1}{2}$

Step 2: Now, perform the multiplication:

$\frac{10 \times 1}{8 \times 2} = \frac{10}{16}$

Next, simplify the fraction $\frac{10}{16}$. The greatest common divisor of 10 and 16 is 2:

Divide the numerator and the denominator by their greatest common divisor:

$\frac{10 \div 2}{16 \div 2} = \frac{5}{8}$

Therefore, the simplified result of the division is $\frac{5}{8}$.

From the answer choices provided, the correct choice is $\frac{5}{8}$.

Thus, the solution to the problem is $\frac{5}{8}$.

To solve the problem $\frac{12}{16} : 3$, we will follow these steps:

• Step 1: Restate the division as a multiplication by using the reciprocal of the divisor. The reciprocal of 3 is $\frac{1}{3}$.
• Step 2: Multiply the fraction by the reciprocal of the divisor: $\frac{12}{16} \times \frac{1}{3}$.
• Step 3: Multiply the numerators: $12 \times 1 = 12$.
• Step 4: Multiply the denominators: $16 \times 3 = 48$.
• Step 5: Simplify the resulting fraction $\frac{12}{48}$.
• Step 6: Both the numerator and the denominator can be divided by 12, which is the greatest common divisor: $\frac{12 \div 12}{48 \div 12} = \frac{1}{4}$.

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