Four numbers are in proportion. The sum of the squares of the four numbers is 50 and the sum of the means is 5. The ratio of first two terms is 1:3. What is the average of the four numbers?

Options

A. 2
B. 3
C. 5
D. 6

Answer

To solve the problem step by step, we will define the four numbers in proportion and use the given conditions to find their values.

### Step 1: Define the Variables
Let the four numbers be \( A, B, C, D \). Since they are in proportion, we can express them as:
- \( A = k \)
- \( B = 3k \) (since the ratio of \( A \) to \( B \) is \( 1:3 \))
- \( C = m \)
- \( D = n \)

### Step 2: Use the Given Conditions
1. The sum of the squares of the four numbers is given as:
\[ A^2 + B^2 + C^2 + D^2 = 50 \]
Substituting the values of \( A \) and \( B \):
\[ k^2 + (3k)^2 + m^2 + n^2 = 50 \]
This simplifies to:
\[ k^2 + 9k^2 + m^2 + n^2 = 50 \]
\[ 10k^2 + m^2 + n^2 = 50 \quad \text{(Equation 1)} \]

2. The sum of the means \( B + C \) is given as:
\[ B + C = 5 \]
Substituting the values of \( B \) and \( C \):
\[ 3k + m = 5 \quad \text{(Equation 2)} \]

### Step 3: Solve for \( m \)
From Equation 2, we can express \( m \) in terms of \( k \):
\[ m = 5 - 3k \]

### Step 4: Substitute \( m \) into Equation 1
Now substitute \( m \) into Equation 1:
\[ 10k^2 + (5 - 3k)^2 + n^2 = 50 \]
Expanding \( (5 - 3k)^2 \):
\[ (5 - 3k)^2 = 25 - 30k + 9k^2 \]
Thus, we have:
\[ 10k^2 + 25 - 30k + 9k^2 + n^2 = 50 \]
Combining like terms:
\[ 19k^2 - 30k + n^2 + 25 = 50 \]
\[ 19k^2 - 30k + n^2 = 25 \quad \text{(Equation 3)} \]

### Step 5: Solve for \( n^2 \)
Rearranging Equation 3 gives:
\[ n^2 = 25 - 19k^2 + 30k \]

### Step 6: Find Possible Values of \( k \)
Since \( n^2 \) must be non-negative, we set up the inequality:
\[ 25 - 19k^2 + 30k \geq 0 \]
This is a quadratic inequality in \( k \). To find the roots, we can use the quadratic formula:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = -19, b = 30, c = -25 \):
\[ k = \frac{-30 \pm \sqrt{30^2 - 4 \cdot (-19) \cdot (-25)}}{2 \cdot (-19)} \]
Calculating the discriminant:
\[ 30^2 - 4 \cdot 19 \cdot 25 = 900 - 1900 = -1000 \quad \text{(no real roots)} \]

### Step 7: Calculate \( D \)
Since \( n^2 \) must be non-negative, we can try specific values for \( k \) to find valid integers for \( A, B, C, D \).

Assuming \( k = 1 \):
- \( A = 1 \)
- \( B = 3 \)
- \( m = 5 - 3(1) = 2 \)
- Substitute \( A, B, C \) into the square equation:
\[ 1^2 + 3^2 + 2^2 + D^2 = 50 \]
\[ 1 + 9 + 4 + D^2 = 50 \]
\[ D^2 = 50 - 14 = 36 \implies D = 6 \]

### Step 8: Calculate the Average
Now we have:
- \( A = 1 \)
- \( B = 3 \)
- \( C = 2 \)
- \( D = 6 \)

The average of the four numbers is:
\[ \text{Average} = \frac{A + B + C + D}{4} = \frac{1 + 3 + 2 + 6}{4} = \frac{12}{4} = 3 \]

### Final Answer
The average of the four numbers is \( 3 \).