Determine the Unknown: Simplify the Fraction Ratio 7x/3z : 4m/3n

To solve this problem, we'll follow these steps:

• Step 1: Write the division problem in terms of multiplication by using the reciprocal.
• Step 2: Multiply the given fractions.
• Step 3: Simplify the resulting expression.

Now, let's work through each step:
Step 1: The problem $\frac{7x}{3z} : \frac{4m}{3n}$ can be rewritten as $\frac{7x}{3z} \times \frac{3n}{4m}$ by taking the reciprocal of $\frac{4m}{3n}$.

Step 2: Multiply the numerators and the denominators:
$\frac{7x \times 3n}{3z \times 4m} = \frac{21xn}{12zm}.$

Step 3: Simplify the fraction by identifying any common factors:
The greatest common divisor of 21 and 12 is 3, so we can divide both by 3:
$\frac{21xn}{12zm} = \frac{7xn}{4zm}.$

The fraction simplifies further by expressing as a mixed number, if needed, to match the format of the answer choices.
The result is $\frac{7}{4} \frac{xn}{zm}$, which can be expressed as the mixed number $1\frac{3}{4} \frac{xn}{zm}$.

Therefore, the solution to the problem is $1\frac{3}{4} \frac{xn}{zm}$.