Dividing Mixed Numbers and Fractions - Examples, Exercises and Solutions

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 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'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, we'll proceed as follows:

• Step 1: Simplify the fraction $\frac{6}{8}$.
• Step 2: Use the formula for dividing by a fraction by multiplying by its reciprocal.
• Step 3: Simplify the resulting fraction or convert it to a mixed number.

Let's work through these steps:

Step 1: Simplify $\frac{6}{8}$.
$\frac{6}{8}$ simplifies to $\frac{3}{4}$ by dividing the numerator and the denominator by 2 (the greatest common divisor).

Step 2: Find the reciprocal of $\frac{3}{4}$ and multiply it by 4.
The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.
So, $4 \div \frac{3}{4} = 4 \times \frac{4}{3} = \frac{16}{3}$.

Step 3: Simplify $\frac{16}{3}$ to a mixed number.
$\frac{16}{3}$ can be expressed as $5\frac{1}{3}$ since 16 divided by 3 is 5 with a remainder of 1.

Therefore, the solution to the problem is $5\frac{1}{3}$.

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 $\frac{6}{7} \div 2$, 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:

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:

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

To solve this problem, we will follow these steps:

• Step 1: Rewrite the division as multiplication by the reciprocal of the fraction.
• Step 2: Perform the multiplication calculation.

Now, let's work through each step:
Step 1: The given problem is $7:\frac{7}{8}$, which means $7 \div \frac{7}{8}$.
Instead of dividing, multiply by the reciprocal:
$7 \div \frac{7}{8} = 7 \times \frac{8}{7}$.

Step 2: Perform the multiplication:
$7 \times \frac{8}{7} = \frac{7 \times 8}{7}$.
The $7$ in the numerator and denominator cancel each other out, resulting in:
$\frac{56}{7} = 8$.

Therefore, the solution to the problem is $8$.

We need to find the value of $3:\frac{2}{3}$, which means dividing 3 by $\frac{2}{3}$.

To solve this, follow these steps:

• Step 1: Find the reciprocal of $\frac{2}{3}$. The reciprocal is obtained by swapping the numerator and the denominator, thus the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
• Step 2: Multiply 3 by the reciprocal $\frac{3}{2}$.
• Step 3: Perform the multiplication: $3 \times \frac{3}{2}$.

Let's execute these steps:

Step 2: Since multiplying a whole number by a fraction gives:

$3 \times \frac{3}{2} = \frac{3 \times 3}{2} = \frac{9}{2}$

Step 3: Convert the improper fraction $\frac{9}{2}$ to a mixed number:

Divide 9 by 2 which gives 4 as the quotient and 1 as the remainder. Thus, the mixed number is $4\frac{1}{2}$.

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

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 this problem, we will follow these steps:

• Step 1: Find the reciprocal of the fraction $\frac{2}{5}$.
• Step 2: Multiply the whole number 2 by this reciprocal.
• Step 3: Simplify the result, if necessary.

Now, let's work through each step:
Step 1: The reciprocal of the fraction $\frac{2}{5}$ is $\frac{5}{2}$.
Step 2: Multiply the whole number 2 by $\frac{5}{2}$:

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

Step 3: Simplify $\frac{10}{2}$:
Divide 10 by 2, which gives us 5.

Therefore, the solution to the problem is $5$.

To solve this problem, let's divide $1$ by $\frac{3}{4}$. The solution involves converting the division into a multiplication:

• Step 1: Recognize $\,1:\frac{3}{4}\,$ as the division $\frac{1}{\frac{3}{4}}$.

• Step 2: Convert division into multiplication: $\frac{1}{\frac{3}{4}} = 1 \times \frac{4}{3}$.

• Step 3: Compute the multiplication: $1 \times \frac{4}{3} = \frac{4}{3}$.

• Step 4: Convert $\frac{4}{3}$ into a mixed number: $1\frac{1}{3}$.

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

The correct answer is $(1 \frac{1}{3})$.