If x is added to each of 14, 12, 34 and 30, the numbers so obtained, in this order, are in proportion. What is the value of `sqrt(12x + 9?)`

Options

A. 13
B. 8
C. 9
D. 11

Answer

To solve the problem step by step, we need to find the value of \( \sqrt{12x + 9} \) given that when \( x \) is added to each of the numbers 14, 12, 34, and 30, they are in proportion.

### Step 1: Set up the proportion
When \( x \) is added to each number, we have:
\[ \frac{14 + x}{12 + x} = \frac{34 + x}{30 + x} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ (14 + x)(30 + x) = (12 + x)(34 + x) \]

### Step 3: Expand both sides
Expanding both sides:
\[ 14 \cdot 30 + 14x + 30x + x^2 = 12 \cdot 34 + 12x + 34x + x^2 \]
Calculating the constants:
\[ 420 + 44x + x^2 = 408 + 46x + x^2 \]

### Step 4: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 420 + 44x = 408 + 46x \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 420 - 408 = 46x - 44x \]
\[ 12 = 2x \]

### Step 6: Solve for \( x \)
Dividing both sides by 2:
\[ x = 6 \]

### Step 7: Substitute \( x \) into \( \sqrt{12x + 9} \)
Now we substitute \( x \) back into the expression:
\[ \sqrt{12x + 9} = \sqrt{12 \cdot 6 + 9} \]
Calculating inside the square root:
\[ = \sqrt{72 + 9} = \sqrt{81} \]

### Step 8: Final calculation
Calculating the square root:
\[ \sqrt{81} = 9 \]

Thus, the value of \( \sqrt{12x + 9} \) is \( 9 \).

---