If x is subtracted from each of 52, 47, 20 and 19. the numbers so obtained in this order are in proportion. What is the mean proportional between ( x + 13) and (x - 3)?

Options

A. 12
B. 10
C. 15
D. 9

Answer

To solve the problem, we will follow these steps:

### Step 1: Set Up the Proportion
We start with the numbers obtained by subtracting \( x \) from 52, 47, 20, and 19. The numbers are:
- \( 52 - x \)
- \( 47 - x \)
- \( 20 - x \)
- \( 19 - x \)

According to the problem, these numbers are in proportion. This means:
\[ \frac{52 - x}{47 - x} = \frac{20 - x}{19 - x} \]

### Step 2: Cross Multiply
To eliminate the fractions, we cross-multiply:
\[ (52 - x)(19 - x) = (47 - x)(20 - x) \]

### Step 3: Expand Both Sides
Now, we will expand both sides of the equation:
- Left Side:
\[ 52 \cdot 19 - 52x - 19x + x^2 = 988 - 71x + x^2 \]
- Right Side:
\[ 47 \cdot 20 - 47x - 20x + x^2 = 940 - 67x + x^2 \]

### Step 4: Set the Equation
Now we set the expanded forms equal to each other:
\[ 988 - 71x + x^2 = 940 - 67x + x^2 \]

### Step 5: Simplify the Equation
We can cancel \( x^2 \) from both sides:
\[ 988 - 71x = 940 - 67x \]

Now, we will move all terms involving \( x \) to one side and constant terms to the other side:
\[ 988 - 940 = 71x - 67x \]
\[ 48 = 4x \]

### Step 6: Solve for \( x \)
Now, we divide both sides by 4:
\[ x = 12 \]

### Step 7: Find the Mean Proportional
Next, we need to find the mean proportional between \( x + 13 \) and \( x - 3 \):
- Calculate \( x + 13 \):
\[ x + 13 = 12 + 13 = 25 \]
- Calculate \( x - 3 \):
\[ x - 3 = 12 - 3 = 9 \]

The mean proportional \( m \) between two numbers \( a \) and \( b \) is given by:
\[ m = \sqrt{a \cdot b} \]
Thus, we calculate:
\[ m = \sqrt{25 \cdot 9} = \sqrt{225} = 15 \]

### Final Answer
The mean proportional between \( x + 13 \) and \( x - 3 \) is \( 15 \).

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