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-2) and (2x + 8)`?

Options

A. 16
B. 12
C. 10
D. 14

Answer

To solve the problem step by step, we need to find the value of \( x \) such that when \( x \) is subtracted from each of the numbers 21, 22, 60, and 64, the resulting numbers are in proportion. Then, we will find the mean proportional between \( (x - 2) \) and \( (2x + 8) \).

### Step 1: Set up the proportion
When \( x \) is subtracted from each number, we have:
- First number: \( 21 - x \)
- Second number: \( 22 - x \)
- Third number: \( 60 - x \)
- Fourth number: \( 64 - x \)

These numbers are in proportion, which means:
\[ \frac{21 - x}{22 - x} = \frac{60 - x}{64 - x} \]

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

### Step 3: Expand both sides
Expanding both sides:
- Left side:
\[ 21 \cdot 64 - 21x - 64x + x^2 = 1344 - 85x + x^2 \]
- Right side:
\[ 22 \cdot 60 - 22x - 60x + x^2 = 1320 - 82x + x^2 \]

### Step 4: Set the equation
Now we set the two sides equal:
\[ 1344 - 85x + x^2 = 1320 - 82x + x^2 \]

### Step 5: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 1344 - 85x = 1320 - 82x \]
Now, simplify:
\[ 1344 - 1320 = 85x - 82x \]
\[ 24 = 3x \]

### Step 6: Solve for \( x \)
Dividing both sides by 3 gives:
\[ x = 8 \]

### Step 7: Find the mean proportional
Now we need to find the mean proportional between \( (x - 2) \) and \( (2x + 8) \):
- Calculate \( x - 2 \):
\[ x - 2 = 8 - 2 = 6 \]
- Calculate \( 2x + 8 \):
\[ 2x + 8 = 2(8) + 8 = 16 + 8 = 24 \]

### Step 8: Mean proportional formula
The mean proportional \( m \) between two numbers \( a \) and \( b \) is given by:
\[ m = \sqrt{a \cdot b} \]
So, we calculate:
\[ m = \sqrt{6 \cdot 24} = \sqrt{144} = 12 \]

### Final Answer
The mean proportional between \( (x - 2) \) and \( (2x + 8) \) is \( 12 \).

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