When the number x is subtracted from each of the numbers 8, 16, and 40, the three numbers that result form a geometric progression. What is the value of x ?

Options

A. 3
B. 4
C. 6
D. 12

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Set up the equation based on the problem statement
We need to find the value of \( x \) such that when \( x \) is subtracted from each of the numbers 8, 16, and 40, the resulting numbers form a geometric progression (GP).

The resulting numbers will be:
- First term: \( 8 - x \)
- Second term: \( 16 - x \)
- Third term: \( 40 - x \)

### Step 2: Use the property of geometric progression
For three numbers \( a, b, c \) to be in geometric progression, the following condition must hold:
\[ \frac{b}{a} = \frac{c}{b} \]
Applying this to our terms:
\[ \frac{16 - x}{8 - x} = \frac{40 - x}{16 - x} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (16 - x)^2 = (40 - x)(8 - x) \]

### Step 4: Expand both sides
Now we will expand both sides of the equation:
- Left side:
\[ (16 - x)^2 = 256 - 32x + x^2 \]
- Right side:
\[ (40 - x)(8 - x) = 320 - 40x - 8x + x^2 = 320 - 48x + x^2 \]

### Step 5: Set the equation
Now we can set the expanded forms equal to each other:
\[ 256 - 32x + x^2 = 320 - 48x + x^2 \]

### Step 6: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 256 - 32x = 320 - 48x \]

### Step 7: Rearrange the equation
Now, let's rearrange the equation to isolate \( x \):
\[ 48x - 32x = 320 - 256 \]
\[ 16x = 64 \]

### Step 8: Solve for \( x \)
Now, divide both sides by 16:
\[ x = \frac{64}{16} = 4 \]

### Final Answer
Thus, the value of \( x \) is \( 4 \).

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