The ratio of two positive numbers is `3 :4`. The sum of their squares is 400. What is the sum of the numbers ?

Options

A. 28
B. 22
C. 24
D. 26

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the numbers based on the ratio
Let the two numbers be represented as:
- First number = 3x
- Second number = 4x

### Step 2: Write the equation for the sum of their squares
According to the problem, the sum of their squares is given as:
\[ (3x)^2 + (4x)^2 = 400 \]

### Step 3: Expand the squares
Now we will expand the squares:
\[ 9x^2 + 16x^2 = 400 \]

### Step 4: Combine like terms
Combine the terms on the left side:
\[ 25x^2 = 400 \]

### Step 5: Solve for x
Now, divide both sides by 25:
\[ x^2 = \frac{400}{25} \]
\[ x^2 = 16 \]
Taking the square root of both sides gives:
\[ x = 4 \]

### Step 6: Find the actual numbers
Now we can find the two numbers using the value of x:
- First number = 3x = 3(4) = 12
- Second number = 4x = 4(4) = 16

### Step 7: Calculate the sum of the numbers
Finally, we can find the sum of the two numbers:
\[ 12 + 16 = 28 \]

### Final Answer
The sum of the numbers is **28**.

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