When x is subtracted from each of 21, 22, 60 and 64, the numbers so obtained in this order are in proportion. What is the mean proportional between (x+4) and `((x)/(2)-1)?`

Options

A. 8
B. 5
C. 6
D. 12

Answer

To solve the problem step by step, we need to determine the value of \( x \) such that when it is subtracted from 21, 22, 60, and 64, the resulting numbers are in proportion. Then, we will find the mean proportional between \( (x + 4) \) and \( \left(\frac{x}{2} - 1\right) \).

### Step 1: Set up the proportion
When \( x \) is subtracted from each of the numbers, we have:
- \( 21 - x \)
- \( 22 - x \)
- \( 60 - x \)
- \( 64 - x \)

According to the property of proportions, we can set up the equation:
\[ \frac{21 - x}{22 - x} = \frac{60 - x}{64 - x} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ (21 - x)(64 - x) = (22 - x)(60 - x) \]

### Step 3: Expand both sides
Expanding both sides, we get:
\[ 21 \cdot 64 - 21x - 64x + x^2 = 22 \cdot 60 - 22x - 60x + x^2 \]

This simplifies to:
\[ 1344 - 85x + x^2 = 1320 - 82x + x^2 \]

### Step 4: Simplify the equation
We can eliminate \( x^2 \) from both sides:
\[ 1344 - 85x = 1320 - 82x \]

Rearranging gives:
\[ 1344 - 1320 = 85x - 82x \]
\[ 24 = 3x \]

### Step 5: Solve for \( x \)
Dividing both sides by 3:
\[ x = 8 \]

### Step 6: Find the mean proportional
Now we need to find the mean proportional between \( (x + 4) \) and \( \left(\frac{x}{2} - 1\right) \).

First, substitute \( x = 8 \):
- \( x + 4 = 8 + 4 = 12 \)
- \( \frac{x}{2} - 1 = \frac{8}{2} - 1 = 4 - 1 = 3 \)

### Step 7: Calculate the mean proportional
The mean proportional \( m \) between two numbers \( a \) and \( b \) is given by:
\[ m = \sqrt{a \cdot b} \]
Thus:
\[ m = \sqrt{12 \cdot 3} = \sqrt{36} = 6 \]

### Final Answer
The mean proportional between \( (x + 4) \) and \( \left(\frac{x}{2} - 1\right) \) is \( 6 \).

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