The average of four consecutive odd natural numbers is eight less than the average of three consecutive even natural numbers. If the sum of these three even numbers is equal to the sum of above four odd numbers, then the average of four original odd numbers is:

Options

A. 24
B. 18
C. 32
D. 36

Answer

To solve the problem step by step, let's denote the four consecutive odd natural numbers as \( x, x+2, x+4, x+6 \) and the three consecutive even natural numbers as \( y, y+2, y+4 \).

### Step 1: Find the average of the four odd numbers
The average of the four odd numbers can be calculated as follows:

\[ \text{Average of odd numbers} = \frac{x + (x+2) + (x+4) + (x+6)}{4} = \frac{4x + 12}{4} = x + 3 \]

### Step 2: Find the average of the three even numbers
The average of the three even numbers is:

\[ \text{Average of even numbers} = \frac{y + (y+2) + (y+4)}{3} = \frac{3y + 6}{3} = y + 2 \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the average of the four odd numbers is eight less than the average of the three even numbers. Therefore, we can set up the following equation:

\[ x + 3 = (y + 2) - 8 \]

This simplifies to:

\[ x + 3 = y - 6 \]

Rearranging gives us:

\[ x - y = -9 \quad \text{(Equation 1)} \]

### Step 4: Set up the equation for the sums
The problem also states that the sum of the three even numbers is equal to the sum of the four odd numbers. Thus, we can write:

\[ (y + (y+2) + (y+4)) = (x + (x+2) + (x+4) + (x+6)) \]

This simplifies to:

\[ 3y + 6 = 4x + 12 \]

Rearranging gives us:

\[ 3y - 4x = 6 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have a system of equations:

1. \( x - y = -9 \)
2. \( 3y - 4x = 6 \)

From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = x + 9 \]

Substituting this into Equation 2:

\[ 3(x + 9) - 4x = 6 \]

Expanding gives:

\[ 3x + 27 - 4x = 6 \]

Combining like terms results in:

\[ -x + 27 = 6 \]

Solving for \( x \):

\[ -x = 6 - 27 \implies -x = -21 \implies x = 21 \]

### Step 6: Find \( y \)
Now substituting \( x = 21 \) back into Equation 1 to find \( y \):

\[ y = 21 + 9 = 30 \]

### Step 7: Calculate the average of the four odd numbers
Now we can find the average of the four odd numbers:

\[ \text{Average of odd numbers} = x + 3 = 21 + 3 = 24 \]

Thus, the average of the four original odd numbers is:

\[ \boxed{24} \]