When x is subtracted from each of 24,33 , 20 and 26, the number so obtained in this order are in proportion . What is the mean proportional between `(2x + 1)` and `(5x - 11)`?

Options

A. 20
B. 42
C. 30
D. 35

Answer

To solve the problem, we will follow these steps:

### Step 1: Set up the equation for proportions
We know that when \( x \) is subtracted from each of the numbers 24, 33, 20, and 26, the resulting numbers are in proportion. This can be expressed as:
\[ \frac{24 - x}{33 - x} = \frac{20 - x}{26 - x} \]

### Step 2: Cross-multiply the proportions
Cross-multiplying gives us:
\[ (24 - x)(26 - x) = (33 - x)(20 - x) \]

### Step 3: Expand both sides
Now, we will expand both sides of the equation:
\[ 24 \times 26 - 24x - 26x + x^2 = 33 \times 20 - 33x - 20x + x^2 \]
Calculating the constants:
\[ 624 - 50x + x^2 = 660 - 53x + x^2 \]

### Step 4: Simplify the equation
Since \( x^2 \) appears on both sides, we can cancel it out:
\[ 624 - 50x = 660 - 53x \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 53x - 50x = 660 - 624 \]
\[ 3x = 36 \]

### Step 6: Solve for \( x \)
Dividing both sides by 3:
\[ x = 12 \]

### Step 7: Find the mean proportional
Now, we need to find the mean proportional between \( 2x + 1 \) and \( 5x - 11 \):
First, substitute \( x = 12 \):
\[ 2x + 1 = 2(12) + 1 = 24 + 1 = 25 \]
\[ 5x - 11 = 5(12) - 11 = 60 - 11 = 49 \]

### Step 8: Calculate the mean proportional
The mean proportional \( m \) between two numbers \( a \) and \( b \) is given by:
\[ m = \sqrt{a \cdot b} \]
Thus,
\[ m = \sqrt{25 \cdot 49} = \sqrt{1225} = 35 \]

### Final Answer
The mean proportional between \( 2x + 1 \) and \( 5x - 11 \) is \( \boxed{35} \).