Mixed Numbers

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: $4\frac{1}{5}$.