If `(a)/(7) = (2b)/(5) = (3a-4b)/(k)`, the value of k is …….

Options

A. 2
B. 5
C. 11
D. 12

Answer

To solve the equation \(\frac{a}{7} = \frac{2b}{5} = \frac{3a - 4b}{k}\), we will follow these steps:

### Step 1: Set the first two expressions equal to each other
We start with the first two parts of the equation:
\[ \frac{a}{7} = \frac{2b}{5} \]

### Step 2: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 5a = 14b \]
From this, we can express \(b\) in terms of \(a\):
\[ b = \frac{5a}{14} \]

### Step 3: Substitute \(b\) into the third expression
Now we substitute \(b\) into the third expression \(\frac{3a - 4b}{k}\):
\[ \frac{3a - 4\left(\frac{5a}{14}\right)}{k} \]
This simplifies to:
\[ \frac{3a - \frac{20a}{14}}{k} = \frac{3a - \frac{10a}{7}}{k} \]

### Step 4: Find a common denominator for the terms in the numerator
To combine the terms in the numerator, we convert \(3a\) to have a denominator of 7:
\[ 3a = \frac{21a}{7} \]
Thus, we have:
\[ \frac{\frac{21a}{7} - \frac{10a}{7}}{k} = \frac{\frac{11a}{7}}{k} \]

### Step 5: Set the two expressions equal to each other
Now we set this equal to the first expression:
\[ \frac{11a}{7k} = \frac{a}{7} \]

### Step 6: Cross-multiply again
Cross-multiplying gives us:
\[ 11a = ak \]

### Step 7: Solve for \(k\)
Assuming \(a \neq 0\), we can divide both sides by \(a\):
\[ k = 11 \]

### Final Answer
Thus, the value of \(k\) is:
\[ \boxed{11} \]

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