Given:

Average of twelve numbers = 39

Average of last five numbers = 35

Average of first four numbers = 40

Fifth number = sixth number - 6

Fifth number = seventh number + 5

Formula used:

Average = (Sum of values)/(Number of values)

Calculations:

Let twelve numbers be x₁, x₂, x₃, x₄, x₅, x₆, x₇, x8, x9, x10, x11, x₁₂.

x5 = x6 - 6      ----(i)

x5 = x7 + 5      ----(ii)

Average of first four numbers,

(x1 + x2 + x3 + x₄)/4 = 40

⇒ (x1 + x2 + x3 + x4) = 160

Average of last five numbers,

(x₈ + x₉ + x₁₀ + x₁₁ + x₁₂)/5 = 35

⇒ (x8 + x9 + x10 + x11 + x12) = 175

Average of twelve numbers,

(x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12)/12 = 39

⇒ (x1 + x2 + x3 + x4 + ​x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12) = 468

⇒ (160 + ​x5 + x6 + x7 + 175) = 468

⇒ ​x5 + x6 + x7 = 468 - 160 - 175

⇒ ​x5 + x6 + x7 = 133      ----(iii)

Putting value of x₅ from equation (i),

⇒ x6 - 6 + x6 + x7 = 133

⇒ 2x6 + x7 = 139

⇒ x7 = 139 - 2x6      ----(iv)

Putting value of x5 from equation (ii),

x7 + 5 + x6 + x7 = 133

⇒ 2x7 + x6 = 128

⇒ 2(139 - 2x6) + x6 = 128

⇒ 278 - 4x₆ + x₆ = 128

⇒ 3x₆ = 150

⇒ x₆ = 50

Putting the value of x₆ in equation (iv),

x7 = 139 - 2x6

⇒ x7 = 139 - 2(50)

⇒ x7 = 139 - 100

⇒ x7 = 39

Average of x₆ and x₇ = (50 + 39)/2

⇒ 89/2

⇒ 44.5

∴  The average of the sixth and seventh numbers is 44.5