If x is subtracted from each of the numbers 20, 37, 54 and 105, then the numbers so obtained in this order are in proportion. What is the mean proportional between (7x - 5) and (x + 1)?

Options

A. 8
B. 6
C. 4
D. 9

Answer

To solve the problem step by step, we need to follow these instructions:

### Step 1: Set up the proportion
We are given four numbers: 20, 37, 54, and 105. If we subtract \( x \) from each of these numbers, they will be in proportion. This can be expressed as:
\[ \frac{20 - x}{37 - x} = \frac{54 - x}{105 - x} \]

### Step 2: Cross-multiply
To eliminate the fractions, we will cross-multiply:
\[ (20 - x)(105 - x) = (37 - x)(54 - x) \]

### Step 3: Expand both sides
Now we will expand both sides of the equation:
- Left side:
\[ 20 \cdot 105 - 20x - 105x + x^2 = 2100 - 125x + x^2 \]
- Right side:
\[ 37 \cdot 54 - 37x - 54x + x^2 = 1998 - 91x + x^2 \]

### Step 4: Set the equation
Now we can set the expanded forms equal to each other:
\[ 2100 - 125x + x^2 = 1998 - 91x + x^2 \]

### Step 5: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 2100 - 125x = 1998 - 91x \]

### Step 6: Rearrange the equation
Now, let's move all terms involving \( x \) to one side and constant terms to the other side:
\[ 2100 - 1998 = 125x - 91x \]
\[ 102 = 34x \]

### Step 7: Solve for \( x \)
Now, divide both sides by 34:
\[ x = \frac{102}{34} = 3 \]

### Step 8: Find the mean proportional
Now we need to find the mean proportional between \( 7x - 5 \) and \( x + 1 \):
- Substitute \( x = 3 \):
\[ 7x - 5 = 7(3) - 5 = 21 - 5 = 16 \]
\[ x + 1 = 3 + 1 = 4 \]

### Step 9: Calculate the mean proportional
The mean proportional \( m \) between two numbers \( a \) and \( b \) is given by:
\[ m = \sqrt{a \cdot b} \]
Thus,
\[ m = \sqrt{16 \cdot 4} = \sqrt{64} = 8 \]

### Final Answer
The mean proportional between \( 7x - 5 \) and \( x + 1 \) is **8**.