The ratio of three positive numbers is `2 : 3 : 5` and the sum of their squares is 608. The three numbers are

Options

A. 2. 3, 5
B. 10, 15. 25
C. 8,12,20
D. 4,6,10

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables based on the ratio
Given the ratio of the three numbers is \(2 : 3 : 5\), we can represent these numbers as:
- First number \(A = 2x\)
- Second number \(B = 3x\)
- Third number \(C = 5x\)

### Step 2: Write the equation for the sum of the squares
According to the problem, the sum of the squares of these numbers is equal to 608. Therefore, we can write the equation:
\[ A^2 + B^2 + C^2 = 608 \]
Substituting the values of \(A\), \(B\), and \(C\):
\[ (2x)^2 + (3x)^2 + (5x)^2 = 608 \]

### Step 3: Expand the squares
Now, we will expand the squares:
\[ 4x^2 + 9x^2 + 25x^2 = 608 \]

### Step 4: Combine like terms
Combine the terms on the left side:
\[ (4 + 9 + 25)x^2 = 608 \]
\[ 38x^2 = 608 \]

### Step 5: Solve for \(x^2\)
Now, divide both sides by 38 to isolate \(x^2\):
\[ x^2 = \frac{608}{38} \]
\[ x^2 = 16 \]

### Step 6: Solve for \(x\)
Taking the square root of both sides gives us:
\[ x = \sqrt{16} = 4 \]

### Step 7: Calculate the three numbers
Now that we have the value of \(x\), we can find the three numbers:
- First number \(A = 2x = 2 \times 4 = 8\)
- Second number \(B = 3x = 3 \times 4 = 12\)
- Third number \(C = 5x = 5 \times 4 = 20\)

### Conclusion
The three numbers are \(8\), \(12\), and \(20\).

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