There are 4 consecutive odd numbers (x₁, x₂, x₃, and x₄) and three consecutive even numbers (y₁, y₂, and y₃). The average of the odd numbers is 6 less than the average of the even numbers. If the sum of the three even numbers is 16 less than the sum of the four odd numbers, find the average of x₁, x₂, x₃, and x₄.

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Accepted Answer

According to the given information:  
Average of odd numbers = Average of even numbers - 6  
   $$\frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{y_1 + y_2 + y_3}{3} - 6$$

Rearranging:  
   $$\frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{y_1 + y_2 + y_3 - 18}{3}$$

Cross multiplying:  
   $$3(x_1 + x_2 + x_3 + x_4) = 4(y_1 + y_2 + y_3) - 72$$

Given that the sum of even numbers is 16 less:  
   $$y_1 + y_2 + y_3 = x_1 + x_2 + x_3 + x_4 - 16$$

Therefore:  
   $$x_1 + x_2 + x_3 + x_4 = y_1 + y_2 + y_3 + 16$$

Substituting this into our equation:  
   $$3(y_1 + y_2 + y_3 + 16) = 4(y_1 + y_2 + y_3) - 72$$

Expanding:  
   $$3y_1 + 3y_2 + 3y_3 + 48 = 4y_1 + 4y_2 + 4y_3 - 72$$

Simplifying:  
   $$y_1 + y_2 + y_3 = 120$$

Finding the sum of odd numbers:  
   $$x_1 + x_2 + x_3 + x_4 = y_1 + y_2 + y_3 + 16 = 120 + 16 = 136$$

Therefore, the average of the odd numbers is:  
    $$\frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{136}{4} = 34$$

The average of the four consecutive odd numbers is **34**.

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