Three numbers are in the ratio 2 : 4 : 5. If the sum of their squares is 1620, then what are the numbers?

Options

A. 30, 36, 12
B. 12, 24, 30
C. 12, 30, 20
D. 10, 12, 36

Answer

To solve the problem step by step, we will follow the given information about the ratio and the sum of the squares of the numbers.

### Step 1: Define the numbers based on the ratio
The three numbers are in the ratio 2:4:5. We can express these numbers in terms of a variable \( x \):
- First number: \( 2x \)
- Second number: \( 4x \)
- Third number: \( 5x \)

### Step 2: Write the equation for the sum of their squares
According to the problem, the sum of the squares of these numbers is equal to 1620. Therefore, we can write the equation:
\[ (2x)^2 + (4x)^2 + (5x)^2 = 1620 \]

### Step 3: Expand the squares
Now we will expand the squares in the equation:
\[ 4x^2 + 16x^2 + 25x^2 = 1620 \]

### Step 4: Combine like terms
Next, we combine the terms on the left side:
\[ (4 + 16 + 25)x^2 = 1620 \]
\[ 45x^2 = 1620 \]

### Step 5: Solve for \( x^2 \)
Now, we will isolate \( x^2 \) by dividing both sides by 45:
\[ x^2 = \frac{1620}{45} \]
Calculating the right side:
\[ x^2 = 36 \]

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

### Step 7: Find the actual numbers
Now that we have the value of \( x \), we can find the three numbers:
- First number: \( 2x = 2 \times 6 = 12 \)
- Second number: \( 4x = 4 \times 6 = 24 \)
- Third number: \( 5x = 5 \times 6 = 30 \)

Thus, the three numbers are **12, 24, and 30**.

### Step 8: Verify the solution
To verify, we can check if the sum of the squares equals 1620:
\[ 12^2 + 24^2 + 30^2 = 144 + 576 + 900 = 1620 \]
This confirms our solution is correct.

### Final Answer:
The numbers are **12, 24, and 30**.

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