Average of 9 numbers is 83. The average of first 2 numbers is 74 while the average of next 4 is 92. If the 8th number is 4 more than the seventh and 2 more than the ninth number, find the eight number?

Sure, let's break it down:

1. The average of 9 numbers is 83. This means the total sum of these 9 numbers is 83 * 9 = 747.

2. The average of the first 2 numbers is 74, so the sum of these 2 numbers is 74 * 2 = 148.

3. The average of the next 4 numbers is 92, so the sum of these 4 numbers is 92 * 4 = 368.

4. Now, if we subtract the sum of the first 6 numbers (148 + 368 = 516) from the total sum of the 9 numbers (747), we get 747 - 516 = 231. This is the sum of the last 3 numbers.

5. We know that the 8th number is 4 more than the 7th and 2 more than the 9th. Let's denote the 7th number as x. This means the 8th number is x + 4 and the 9th number is x + 4 - 2 = x + 2.

6. If we add these three numbers together (x + x + 4 + x + 2), we get 3x + 6. We know that this sum is equal to 231 (from step 4), so we can set up the equation 3x + 6 = 231.

7. Solving this equation for x gives us x = (231 - 6) / 3 = 75. This is the 7th number.

8. The 8th number is 4 more than the 7th, so it is 75 + 4 = 79.

So, the 8th number is 79.