Three numbers are such that their sum is 50, product is 3750 and the sum of their reciprocals is `(31)/(150)`. Find the sum of the squares of the three numbers.

Options

A. 2500
B. 1250
C. 950
D. 122

Answer

To find the sum of the squares of the three numbers, we can follow these steps:

### Step 1: Define the Variables
Let the three numbers be \( x, y, z \).

### Step 2: Write Down the Given Information
We are given:
1. \( x + y + z = 50 \) (the sum of the numbers)
2. \( xyz = 3750 \) (the product of the numbers)
3. \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{31}{150} \) (the sum of the reciprocals)

### Step 3: Express the Sum of Reciprocals
The sum of the reciprocals can be rewritten using the formula:
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{xy + yz + zx}{xyz} \]
From the given information, we can substitute:
\[ \frac{xy + yz + zx}{3750} = \frac{31}{150} \]

### Step 4: Solve for \( xy + yz + zx \)
Cross-multiplying gives:
\[ xy + yz + zx = 3750 \times \frac{31}{150} \]
Calculating this:
\[ xy + yz + zx = 3750 \times \frac{31}{150} = 3750 \div 150 \times 31 = 25 \times 31 = 775 \]

### Step 5: Use the Formula for the Sum of Squares
We need to find \( x^2 + y^2 + z^2 \). We can use the identity:
\[ x^2 + y^2 + z^2 = (x + y + z)^2 - 2(xy + yz + zx) \]
Now substituting the known values:
\[ x^2 + y^2 + z^2 = (50)^2 - 2(775) \]
Calculating:
\[ x^2 + y^2 + z^2 = 2500 - 1550 = 950 \]

### Final Answer
The sum of the squares of the three numbers is \( 950 \).

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