If x and y are positive integers and ` ( 3 x )/( y) = 4` then find the minimum value of `( 3 x + 4y) `

Options

A. 24
B. 1
C. 6
D. 8

Answer

To solve the problem, we need to find the minimum value of the expression \(3x + 4y\) given the condition \(\frac{3x}{y} = 4\). Let's break this down step by step.

### Step 1: Rearranging the equation
From the equation \(\frac{3x}{y} = 4\), we can rearrange it to express \(y\) in terms of \(x\):
\[ 3x = 4y \implies y = \frac{3x}{4} \]

### Step 2: Finding integer values for \(y\)
Since \(y\) must be a positive integer, \(\frac{3x}{4}\) must also be an integer. This means that \(3x\) must be divisible by \(4\). Therefore, \(x\) must be a multiple of \(4\) because \(3\) and \(4\) are coprime.

### Step 3: Finding the smallest positive integer \(x\)
The smallest positive integer \(x\) that satisfies this condition is \(4\) (the smallest multiple of \(4\)):
\[ x = 4 \]

### Step 4: Calculating \(y\)
Now we can substitute \(x = 4\) back into the equation for \(y\):
\[ y = \frac{3(4)}{4} = 3 \]

### Step 5: Calculating the expression \(3x + 4y\)
Now that we have both \(x\) and \(y\), we can find the value of the expression \(3x + 4y\):
\[ 3x + 4y = 3(4) + 4(3) = 12 + 12 = 24 \]

### Conclusion
Thus, the minimum value of \(3x + 4y\) is:
\[ \boxed{24} \]