Divide Mixed Number 2 4/7 by Fraction 3/4: Step-by-Step Solution

To solve the problem of dividing the mixed number $2\frac{4}{7}$ by the fraction $\frac{3}{4}$, follow these steps:

• Step 1: Convert the mixed number $2\frac{4}{7}$ into an improper fraction.
• Step 2: Calculate the reciprocal of the divisor, $\frac{3}{4}$.
• Step 3: Multiply the improper fraction by the reciprocal.
• Step 4: Simplify the resulting fraction and convert it to a mixed number if applicable.

Now, let us solve the problem step by step:

Step 1: Convert the mixed number $2\frac{4}{7}$ into an improper fraction.

$2\frac{4}{7} = \frac{2 \times 7 + 4}{7} = \frac{14 + 4}{7} = \frac{18}{7}$

Step 2: Calculate the reciprocal of the divisor $\frac{3}{4}$.

The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.

Step 3: Multiply the improper fraction $\frac{18}{7}$ by the reciprocal $\frac{4}{3}$.

$\frac{18}{7} \times \frac{4}{3} = \frac{18 \times 4}{7 \times 3} = \frac{72}{21}$

Step 4: Simplify $\frac{72}{21}$ and convert it back to a mixed number.

We find the greatest common divisor of 72 and 21, which is 3, and divide both the numerator and the denominator by 3:

$\frac{72}{21} = \frac{72 \div 3}{21 \div 3} = \frac{24}{7}$

Finally, convert $\frac{24}{7}$ back to a mixed number.

$\frac{24}{7} = 3\frac{24 - 21}{7} = 3\frac{3}{7}$

However, the final answer should match one of the given choices, specifically in a typographical form equivalent to one of them:

The equivalent expression of $3\frac{9}{21}$ (from choice 4) when $\frac{3}{7}$ is simplified and matches the correct result:

$\frac{9}{21}$ is indeed $\frac{3}{7}$.

Therefore, the correct answer is $\boxed{3\frac{9}{21}}$, as demonstrated by matching all criteria set by the problem statement.