All Operations in Mixed Fractions: Dividing whole numbers by fractions

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 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 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, let's carry out the following steps:

• Step 1: Recognize that the expression $3 : \frac{5}{6}$ represents division, so it becomes $3 \div \frac{5}{6}$.
• Step 2: Use the rule that dividing by a fraction is the same as multiplying by its reciprocal. Thus, convert this to $3 \times \frac{6}{5}$.
• Step 3: Multiply 3 (which can be written as $\frac{3}{1}$) by $\frac{6}{5}$:
  $\frac{3}{1} \times \frac{6}{5} = \frac{3 \times 6}{1 \times 5} = \frac{18}{5}$.
• Step 4: Convert $\frac{18}{5}$ to a mixed number. Divide 18 by 5:
  - 5 goes into 18 three times with a remainder of 3.
  - Therefore, $\frac{18}{5} = 3\frac{3}{5}$.

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

To solve the problem $4 : \frac{4}{7}$, we'll apply the concept of dividing by a fraction:

Step 1: Convert the division to multiplication by the reciprocal of the fraction. The reciprocal of $\frac{4}{7}$ is $\frac{7}{4}$.

Thus, the expression $4 : \frac{4}{7}$ is equivalent to $4 \times \frac{7}{4}$.

Step 2: Perform the multiplication:

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

Step 3: Multiply the numerators and the denominators:

$\frac{4 \times 7}{1 \times 4} = \frac{28}{4}$

Step 4: Simplify the fraction:

$\frac{28}{4} = 7$

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

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 $(1 \frac{1}{3})$.

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 $4:\frac{3}{5}$, follow these steps:

• Step 1: Convert the division into a multiplication by using the reciprocal of $\frac{3}{5}$. The reciprocal is $\frac{5}{3}$.
• Step 2: Multiply the whole number 4 by $\frac{5}{3}$:
  $4 \times \frac{5}{3} = \frac{4 \times 5}{3} = \frac{20}{3}$
• Step 3: Simplify $\frac{20}{3}$ to a mixed number:
  Perform the division $20 \div 3$, which gives 6 as a whole number and leaves a remainder of 2.
• Thus, $\frac{20}{3}$ as a mixed number is $6\frac{2}{3}$.

Therefore, the solution to the problem is $6\frac{2}{3}$.

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 this problem, we'll perform the division $5 \div \frac{2}{5}$ by converting it into multiplication:

• Step 1: Recognize that dividing by a fraction is the same as multiplying by its reciprocal.
• Step 2: Convert the division into multiplication: $5 \div \frac{2}{5} = 5 \times \frac{5}{2}$.
• Step 3: Multiply the whole number by the reciprocal of the fraction: $5 \times \frac{5}{2} = \frac{25}{2}$.
• Step 4: Convert the improper fraction to a mixed number: $\frac{25}{2} = 12 \frac{1}{2}$.

Through these steps, we find that the solution to the division problem is $12 \frac{1}{2}$.

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.