Given that `(2m+n)/(2n)=(3)/(4)`, which of the following must also be true?

Options

A. `m=(1)/(8)`
B. `m=(3)/(4)`
C. `(m)/(n)=(1)/(4)`
D. `(m)/(n)=(5)/(4)`

Explanation

To solve the equation \(\frac{2m+n}{2n}=\frac{3}{4}\) and determine which of the following must also be true, we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \frac{2m+n}{2n} = \frac{3}{4} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 4(2m + n) = 3(2n) \]

### Step 3: Distribute both sides
Distributing the terms, we get:
\[ 8m + 4n = 6n \]

### Step 4: Rearrange the equation
Now, let's move \(4n\) to the right side:
\[ 8m = 6n - 4n \]
This simplifies to:
\[ 8m = 2n \]

### Step 5: Solve for \(m\) in terms of \(n\)
Now, divide both sides by 8:
\[ m = \frac{2n}{8} \]
This simplifies to:
\[ m = \frac{n}{4} \]

### Step 6: Express \(m\) in terms of \(n\)
We can express this as:
\[ \frac{m}{n} = \frac{1}{4} \]

### Conclusion
Thus, we find that:
\[ \frac{m}{n} = \frac{1}{4} \]
This indicates that option C must be true.

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