Dividing Mixed Numbers and Fractions - Examples, Exercises and Solutions

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 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 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 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'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 $6$.

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 $\frac{1}{8}$.

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 divide the whole number 3 by the fraction $\frac{5}{7}$, we follow these steps:

• Step 1: Identify the reciprocal of the fraction. The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.
• Step 2: Multiply the whole number 3 by this reciprocal.
• Step 3: Perform the multiplication to find the result.

Let's calculate this:
Step 1: The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.
Step 2: Multiply: $3 \times \frac{7}{5} = \frac{3 \times 7}{5} = \frac{21}{5}$.
Step 3: Convert the improper fraction $\frac{21}{5}$ to a mixed number:

• Divide 21 by 5. It goes 4 times with a remainder of 1.
• The quotient is 4, and the remainder is 1. Therefore, $\frac{21}{5} = 4\frac{1}{5}$.

Thus, the solution to $3 : \frac{5}{7}$ is $4\frac{1}{5}$.

The correct choice among the given answers is: $4\frac{1}{5}$.

To solve the expression $2:\frac{2}{3}$, follow these steps:

• Step 1: Rewrite the expression as a division problem:
  This means $2 \div \frac{2}{3}$.
• Step 2: Convert the division to a multiplication by using the reciprocal:
  The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
• Step 3: Multiply by the reciprocal:
  $2 \times \frac{3}{2} = \frac{2 \cdot 3}{2 \cdot 1} = \frac{6}{2} = 3$.

Therefore, the solution to the problem is $3$.

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 $\frac{4}{5}$ 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 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 the problem $3:\frac{3}{4}$, we must perform division of the whole number 3 by the fraction $\frac{3}{4}$. Here are the steps:

• Step 1: Recall the rule for dividing by a fraction. Dividing by $\frac{3}{4}$ is the same as multiplying by its reciprocal, $\frac{4}{3}$.
• Step 2: Rewrite the expression as a multiplication problem: $3 \times \frac{4}{3}$.
• Step 3: Perform the multiplication: $3 \times \frac{4}{3} = \frac{3 \times 4}{3} = \frac{12}{3}$.
• Step 4: Simplify the fraction: $\frac{12}{3} = 4$.

The solution to the division $3:\frac{3}{4}$ is $4$.

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 $\frac{3}{7}$.

To solve this problem, follow these steps:

• Step 1: Identify the problem as dividing the fraction $\frac{3}{4}$ by the whole number 3.
• Step 2: Understand that dividing by 3 is equivalent to multiplying by its reciprocal, which is $\frac{1}{3}$.
• Step 3: Multiply $\frac{3}{4}$ by $\frac{1}{3}$.
• Step 4: Simplify the result if possible.

Let's solve this step-by-step:
Step 1: The given expression is $\frac{3}{4} \div 3$.
Step 2: Rewrite the division as a multiplication using the reciprocal: $\frac{3}{4} \times \frac{1}{3}$.
Step 3: Perform the multiplication: $\frac{3}{4} \times \frac{1}{3} = \frac{3 \times 1}{4 \times 3} = \frac{3}{12}$.
Step 4: Simplify $\frac{3}{12}$ by dividing the numerator and the denominator by their greatest common divisor, which is 3:
$\frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4}$.

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