The Problem
We are given the following information:
- The average of twelve numbers is 39.
- The average of the last five numbers is 35.
- The average of the first four numbers is 40.
- The fifth number is 6 less than the sixth number.
- The fifth number is 5 more than the seventh number.

We need to find the average of the fifth and sixth numbers.

Solution
To solve this problem, we can use the concept of averages and algebraic equations.

Average of Twelve Numbers
We are given that the average of twelve numbers is 39. Let's denote the sum of these twelve numbers as S.

Therefore, we can write the equation:
S/12 = 39

Average of Last Five Numbers
We are given that the average of the last five numbers is 35. Let's denote the sum of these five numbers as L.

Therefore, we can write the equation:
L/5 = 35

Average of First Four Numbers
We are given that the average of the first four numbers is 40. Let's denote the sum of these four numbers as F.

Therefore, we can write the equation:
F/4 = 40

Relationship between Fifth, Sixth, and Seventh Numbers
We are given two relationships between the fifth, sixth, and seventh numbers:
1. The fifth number is 6 less than the sixth number.
2. The fifth number is 5 more than the seventh number.

Let's denote the fifth number as x, the sixth number as y, and the seventh number as z.

From the first relationship, we can write the equation:
x = y - 6

From the second relationship, we can write the equation:
x = z + 5

Now, let's solve these equations to find the values of x, y, and z.

Solving the Equations
From the equation x = y - 6, we can substitute this value of x in the equation x = z + 5:
y - 6 = z + 5

Simplifying this equation, we get:
y - z = 11 ... (Equation 1)

Now, let's solve the equations for the averages.

From the equation S/12 = 39, we can write:
S = 12 * 39
S = 468 ... (Equation 2)

From the equation L/5 = 35, we can write:
L = 5 * 35
L = 175 ... (Equation 3)

From the equation F/4 = 40, we can write:
F = 4 * 40
F = 160 ... (Equation 4)

Finding the Values of x, y, and z
To find the values of x, y, and z, we need to solve the equations.

From Equation 2, we know that the sum of all twelve numbers is 468. Therefore, we can write:
x + y + z + (sum of the other nine numbers) = 468

Since the average of the last five numbers is 35 (Equation 3), the sum of these five numbers is 175. Therefore, we can write:
y + z + (sum of the other three numbers) =