1: solution: suppose that two candles go out at the same time after x hours.

1- 1/5 x =( 1- 1/4 x)×4，

1- 1/ 5 x=4-x，

1- 1 /5 x+x=4-x+x，

1+4/ 5 x=4，

1+4 /5 x- 1=4- 1，

4/ 5 x÷4/ 5 =3÷4/ 5，

x = 3(3/4)；

3(3 /4) hours =3 hours and 45 minutes,

8: 00 +3: 45 = 1 1: 45,

Answer: The candle went out at 1 1: 45 in the afternoon.

2. Solution: When the remaining 1-3/ 5 =2 /5 parts are processed, the work efficiency becomes the original (1+20%)=6 /5.

Then the time used is 5/ 6 of the original processing time, which is less than the original 1/6.

So 10 days in advance is the original time: 10÷ 1 /6 =60 (days);

Then the original planned processing time of this batch of parts is 60÷2 /5 = 150 (days);

These parts * * * are: 15× 150=2250 (pieces).

A: There are 2250 parts in this batch.