For 15 numbers, the overall average is 60. The average of the first 8 of them is 58 and the average of the last 8 is 63. By how much does the 8th number exceed the average of all 15 numbers?

All 15 total $15\times 60=900$. The first 8 total $8\times 58=464$ and the last 8 total $8\times 63=504$; adding those gives 968, which covers all 15 numbers plus the 8th a second time, since the 8th belongs to both blocks. So the 8th number is 968 - 900 = 68, which exceeds the overall average of 60 by 8. 68 is the 8th number itself, not the excess asked for. 5 is the gap between the two block averages. 1 comes from the guess 58 + 63 - 60 = 61, which treats the averages as if they were the totals.