Given:

Average of 12 numbers = 48

Average of first 5 numbers = 45

Average of next 4 numbers = 52

10th number = 11th number – 10

10th number = 12th number + 5

Formula used:

Sum = Average × Number of terms

Calculations:

Total sum of 12 numbers = 48 × 12 = 576

Sum of first 5 numbers = 45 × 5 = 225

Sum of next 4 numbers = 52 × 4 = 208

Sum of 10th, 11th, and 12th numbers = 576 – (225 + 208) = 143

Let the 11th number = x.

Then, 10th number = x – 10 and 12th number = x – 15.

Sum of 10th, 11th, and 12th numbers:

(x – 10) + x + (x – 15) = 143

⇒ 3x – 25 = 143

⇒ 3x = 168

⇒ x = 56

11th number = 56

12th number = 56 – 15 = 41

Average of 11th and 12th numbers = (56 + 41) ÷ 2 = 48.5

∴ The average of the 11th and 12th numbers is 48.5.