Given Information:
- The average of sixteen numbers is 48.
- The average of the first six of these numbers is 45.
- The average of the last seven numbers is 53.
- The seventh and eighth numbers are 3 and 7 greater than the ninth number.

Calculating the Sum of All Numbers:
We know that the average of sixteen numbers is 48. Therefore, the sum of these sixteen numbers can be calculated by multiplying the average by the total count of numbers:
Sum of sixteen numbers = 16 * 48 = 768

Calculating the Sum of the First Six Numbers:
We are given that the average of the first six numbers is 45. Therefore, the sum of the first six numbers can be calculated by multiplying the average by the count of numbers:
Sum of the first six numbers = 6 * 45 = 270

Calculating the Sum of the Last Seven Numbers:
We are given that the average of the last seven numbers is 53. Therefore, the sum of the last seven numbers can be calculated by multiplying the average by the count of numbers:
Sum of the last seven numbers = 7 * 53 = 371

Calculating the Sum of the Seventh and Eighth Numbers:
We are given that the seventh and eighth numbers are 3 and 7 greater than the ninth number. Let's assume the ninth number is x. Therefore, the seventh number is x + 3 and the eighth number is x + 7.
Sum of the seventh and eighth numbers = (x + 3) + (x + 7) = 2x + 10

Calculating the Sum of All Numbers Except the Ninth, Seventh, and Eighth Numbers:
We can calculate the sum of all numbers except the ninth, seventh, and eighth numbers by subtracting the sum of the first six numbers, the sum of the last seven numbers, and the sum of the seventh and eighth numbers from the sum of all sixteen numbers:
Sum of all numbers except the ninth, seventh, and eighth numbers = 768 - 270 - 371 - (2x + 10)

Calculating the Ninth Number:
The average of all sixteen numbers is 48. Therefore, the sum of all sixteen numbers except the ninth, seventh, and eighth numbers should be multiplied by 16 (count of all numbers) and subtracted from the sum of all sixteen numbers to find the value of the ninth number:
16 * (Sum of all numbers except the ninth, seventh, and eighth numbers) = 768 - (2x + 10)

Calculating the Average of the Ninth and Seventh Numbers:
To find the average of the ninth and seventh numbers, we need to calculate the sum of these two numbers and divide it by 2 (count of numbers):
Average of the ninth and seventh numbers = (9th number + 7th number) / 2

Summary:
- The ninth number can be found by solving the equation: 16 * (Sum of all numbers except the ninth, seventh, and eighth numbers) = 768 - (2x + 10).
- The average of the ninth and seventh numbers can be calculated by finding the sum of these two numbers and dividing it by 2.