The sum of the squares of three numbers which are in the ratio 2:3:4 is 725. What are these numbers?

Options

A. 10, 15, 20
B. 14, 21, 28
C. 20, 15, 30
D. 20, 30, 40

Answer

To solve the problem step by step, we will follow the method outlined in the video transcript.

### Step 1: Define the Numbers
Let the three numbers be represented in terms of a variable \( x \). Since the numbers are in the ratio 2:3:4, we can express them as:
- First number = \( 2x \)
- Second number = \( 3x \)
- Third number = \( 4x \)

### Step 2: Set Up the Equation
According to the problem, the sum of the squares of these numbers is equal to 725. Therefore, we can write the equation:
\[ (2x)^2 + (3x)^2 + (4x)^2 = 725 \]

### Step 3: Expand the Squares
Now, we will expand the squares in the equation:
\[ 4x^2 + 9x^2 + 16x^2 = 725 \]

### Step 4: Combine Like Terms
Next, we combine the like terms (the \( x^2 \) terms):
\[ (4 + 9 + 16)x^2 = 725 \]
This simplifies to:
\[ 29x^2 = 725 \]

### Step 5: Solve for \( x^2 \)
Now, we will isolate \( x^2 \) by dividing both sides of the equation by 29:
\[ x^2 = \frac{725}{29} \]
Calculating the right side gives:
\[ x^2 = 25 \]

### Step 6: Solve for \( x \)
Taking the square root of both sides, we find:
\[ x = \sqrt{25} = 5 \]

### Step 7: Find the Numbers
Now that we have the value of \( x \), we can find the actual numbers:
- First number = \( 2x = 2 \times 5 = 10 \)
- Second number = \( 3x = 3 \times 5 = 15 \)
- Third number = \( 4x = 4 \times 5 = 20 \)

### Final Answer
Thus, the three numbers are 10, 15, and 20.

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