Dividing Mixed Numbers and Fractions - Examples, Exercises and Solutions

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

• Step 1: Understand that dividing by 3 is equivalent to multiplying by the reciprocal of 3, which is $\frac{1}{3}$.
• Step 2: Convert the division problem $\frac{1}{3} : 3$ into a multiplication problem $\frac{1}{3} \times \frac{1}{3}$.
• Step 3: Perform the multiplication of fractions:

Using the formula $\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$, we have:

$\frac{1}{3} \times \frac{1}{3} = \frac{1 \cdot 1}{3 \cdot 3} = \frac{1}{9}$.

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

To solve $\frac{1}{2} : 2$, we need to rewrite the division as multiplication by the reciprocal of 2. The reciprocal of 2 is $\frac{1}{2}$.

Thus, the expression becomes:

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

Calculating the multiplication, we have:

$\frac{1}{4}$

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

To solve this problem of dividing a fraction by a whole number, we'll follow these steps:

• Step 1: Change the whole number to a reciprocal fraction.
• Step 2: Multiply the original fraction by the reciprocal.
• Step 3: Simplify the resulting fraction, if necessary.

Now, let's apply these steps:
Step 1: The whole number $3$ is converted to the reciprocal fraction $\frac{1}{3}$.
Step 2: Multiply the fraction $\frac{1}{2}$ by $\frac{1}{3}$:

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

Step 3: The resulting fraction $\frac{1}{6}$ is already in its simplest form.

Therefore, when $\frac{1}{2}$ is divided by $3$, the resulting answer is $\frac{1}{6}$.

To solve this problem, we need to compute $\frac{1}{2} \div 4$. Here are the steps:

• Step 1: Recognize that dividing by 4 is equivalent to multiplying by its reciprocal, $\frac{1}{4}$.
• Step 2: Rewrite the division as a multiplication: $\frac{1}{2} \times \frac{1}{4}$.
• Step 3: Perform the multiplication of fractions by multiplying their numerators and denominators. Thus, $\frac{1 \cdot 1}{2 \cdot 4} = \frac{1}{8}$.

Therefore, the solution to the problem is $\frac{1}{8}$.

To solve the division problem $1 : \frac{1}{4}$, we will follow these steps:

• Step 1: Express the division as a fraction operation: $1 \div \frac{1}{4}$.
• Step 2: Use the invert-and-multiply rule. Find the reciprocal of $\frac{1}{4}$, which is $4$.
• Step 3: Multiply the whole number by the reciprocal: $1 \times 4 = 4$.

Thus, after performing these operations, we find that the result of the division $1 : \frac{1}{4}$ is $4$.

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

• Step 1: Identify the reciprocal of the divisor.
• Step 2: Multiply the dividend by the reciprocal.

Now, let's work through each step:
Step 1: The divisor is $\frac{1}{2}$. The reciprocal of $\frac{1}{2}$ is 2.

Step 2: Multiply the dividend, which is 3, by the reciprocal of the divisor:
$3 \times 2 = 6$

Therefore, the solution to the problem is $6$.

We need to evaluate the expression $1 \div \frac{2}{3}$.

To do this, we use the principle that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, the expression becomes:

$1 \times \frac{3}{2}$.

Next, we multiply the whole number by the reciprocal:

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

To express $\frac{3}{2}$ as a mixed number, we write it as:

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

Thus, the solution to the problem is $1\frac{1}{2}$, which matches choice 3 from the options provided.

To solve this problem of dividing a fraction by a whole number, we will follow these steps:

• Step 1: Convert the division problem into a multiplication by the reciprocal.
• Step 2: Multiply the fraction by the reciprocal of the whole number.
• Step 3: Simplify the resulting fraction, if possible.

Let's work through these steps in detail:

Step 1: Convert the division into a multiplication by the reciprocal.
The given problem is $\frac{2}{3} : 5$. In arithmetic, division by a whole number can be converted into multiplication by its reciprocal. The reciprocal of the whole number 5 is $\frac{1}{5}$. Therefore, the expression becomes:

$\frac{2}{3} \times \frac{1}{5}$.

Step 2: Multiply the fraction by the reciprocal.
We now multiply the numerators and the denominators:

$\frac{2}{3} \times \frac{1}{5} = \frac{2 \times 1}{3 \times 5} = \frac{2}{15}$.

Step 3: Simplify the resulting fraction.
We check if the fraction $\frac{2}{15}$ can be simplified further. Since 2 and 15 have no common divisors besides 1, the fraction is already in its simplest form.

Therefore, the solution to the division problem $\frac{2}{3} : 5$ is $\frac{2}{15}$.

Upon examining the provided answer choices, we confirm that our solution, $\frac{2}{15}$, matches choice number 4.

To solve the problem of dividing the fraction $\frac{4}{5}$ by 2, we can utilize the method of multiplying by the reciprocal. Here’s how you can systematically approach it:

Given the division $\frac{4}{5} : 2$, we first express the division by finding the reciprocal.
Step 1: The reciprocal of 2 is $\frac{1}{2}$.

Step 2: Now, multiply the fraction $\frac{4}{5}$ by $\frac{1}{2}$:

$\frac{4}{5} \times \frac{1}{2} = \frac{4 \times 1}{5 \times 2} = \frac{4}{10}$

Step 3: Simplify the resulting fraction:

The numerator and the denominator have a common factor of 2. Dividing both by 2 gives:

$\frac{4 \div 2}{10 \div 2} = \frac{2}{5}$

Therefore, the solution to the problem $\frac{4}{5} : 2$ is $\frac{2}{5}$.

To solve this problem, follow these steps:

• Step 1: Recognize that dividing by a number is equivalent to multiplying by its reciprocal.
• Step 2: Apply this transformation to the problem, emphasizing the operation change.
• Step 3: Perform the multiplication operation correctly.
• Step 4: Simplify the resulting fraction to simplest terms.

Let's work through it:

Step 1: Start with the expression: $\frac{3}{4} \div 4$.

Step 2: Convert the division by 4 into multiplication by its reciprocal, $\frac{1}{4}$. The expression becomes: $\frac{3}{4} \times \frac{1}{4}$.

Step 3: To multiply fractions, multiply the numerators together and the denominators together:
Numerator: $3 \times 1 = 3$
Denominator: $4 \times 4 = 16$

Therefore, the resulting fraction from the multiplication is $\frac{3}{16}$.

There is no need for additional simplification as $\frac{3}{16}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{3}{16}$.

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

• Step 1: Rewrite the division as a multiplication by the reciprocal.
• Step 2: Perform the multiplication of fractions.
• Step 3: Simplify if necessary and find the solution among the given choices.

Now, let's work through each step:
Step 1: Rewrite the problem $\frac{2}{7} \div 3$ as a multiplication:
$\frac{2}{7} \times \frac{1}{3}$

Step 2: Multiply the numerators and the denominators:
$\frac{2 \times 1}{7 \times 3} = \frac{2}{21}$

Step 3: This fraction is already in its simplest form. Looking at the answer choices, we can conclude the correct answer is $\frac{2}{21}$.

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

To solve the problem of dividing $\frac{4}{7}$ by 5, we will follow these steps:

• Step 1: Understand that dividing by a number is the same as multiplying by its reciprocal.
• Step 2: Convert the division problem into a multiplication problem using the reciprocal.
• Step 3: Perform the multiplication of fractions.

Now, let's implement these steps:

Step 1: We have the fraction $\frac{4}{7}$ and need to divide it by the whole number 5. In terms of fractions, 5 can be written as $\frac{5}{1}$.

Step 2: Change the division into multiplication. This requires us to multiply $\frac{4}{7}$ by the reciprocal of $\frac{5}{1}$, which is $\frac{1}{5}$. Thus, the expression becomes:

$\frac{4}{7} \div 5 = \frac{4}{7} \times \frac{1}{5}$

Step 3: Multiply the fractions. To multiply fractions, multiply the numerators and multiply the denominators:

$\frac{4}{7} \times \frac{1}{5} = \frac{4 \times 1}{7 \times 5} = \frac{4}{35}$

Therefore, the final result of dividing $\frac{4}{7}$ by 5 is $\frac{4}{35}$.

To solve the problem of dividing the fraction $\frac{5}{8}$ by 2, we can follow these steps:

• Step 1: Understand that dividing by a whole number is the same as multiplying by its reciprocal. That means $\frac{5}{8} \div 2$ is equivalent to $\frac{5}{8} \times \frac{1}{2}$.
• Step 2: Perform the multiplication of fractions. Multiply the numerators and the denominators:

$\frac{5}{8} \times \frac{1}{2} = \frac{5 \times 1}{8 \times 2} = \frac{5}{16}$

Therefore, the solution to the problem is $\frac{5}{16}$.

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

• Step 1: Convert the whole number to a fraction and find its reciprocal.
• Step 2: Multiply the given fraction by this reciprocal.
• Step 3: Simplify the result, if possible.

Let's tackle each step:
Step 1: Convert $4$ to a fraction, which is $\frac{4}{1}$, and find the reciprocal giving $\frac{1}{4}$.
Step 2: Multiply $\frac{3}{5}$ by $\frac{1}{4}$ to get $\frac{3}{5} \times \frac{1}{4} = \frac{3 \times 1}{5 \times 4} = \frac{3}{20}$.
Step 3: The fraction $\frac{3}{20}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{3}{20}$.

To solve the problem $\frac{6}{7} \div 2$, we need to remember how to divide a fraction by a whole number:

Step 1: Convert the division problem into a multiplication problem by multiplying by the reciprocal of the whole number. The reciprocal of 2 is $\frac{1}{2}$.

Step 2: Therefore, we rewrite the problem as $\frac{6}{7} \times \frac{1}{2}$.

Step 3: Multiply the numerators together and the denominators together:

$\frac{6 \times 1}{7 \times 2} = \frac{6}{14}$

Step 4: Simplify the fraction $\frac{6}{14}$ by finding the greatest common divisor of 6 and 14, which is 2. Divide both the numerator and the denominator by 2:

$\frac{6 \div 2}{14 \div 2} = \frac{3}{7}$

Therefore, the solution to the problem is $\frac{3}{7}$.