Solution:

We agree that the series is quite difficult to crack and much tricky as well.

1 can be written as one (consist of 3 letters)
2 can be written as two (consist of 3 letters)
3 can be written as five (consist of 5 letters)
4 can be written as four (consist of 4 letters)
5 can be written as five (consist of 5 letters)
6 can be written as six (consist of 3 letters)

Thus the next term in the series is 3.

Solution:

John starts after 3 days, thus Peter has already covered 60 km by then (travelling at 20km a day)
The distance between Peter and John will be increased to 60 + 5 + 1 = 66 km in the next two days.

Since John is following an arithmetic progression in the speed daily, the distance between him and peter will keep on decreasing from the 6th day. It will decrease by 3 km on the first day, 7 km on the next day and so on.

Now John will catch up with Peter after n + 5 days from the start of Peter
Thus
66 = (n/2)*(2*3 + (n-1)*4)

If we solve for n in the above equation, we will get n = 5.5

Thus John will catch up with Peter on the eights day when he starts running after Peter.

Solution:

39! / 41
1*2*3*4....*39 / 41
Rem (1 / 41) + rem (2 / 41) + rem (3 / 41) .... + rem (39 / 41)
= rem ((1 + 2 + 3 + ...39) / 41)
= rem (780 / 41)
= 1

Solution:

Let us assume I bought 100 m of cloth for 100 Rs.
Thus I must be selling the cloth for Rs. 90 (10 percent loss).

But using false measurement, I gained a profit of 15 percent

Thus CP of the sold cloth = 90/1.15 = Rs. 78.26

Thus I must have sold 78.26 meters in place of 100 meters.

Solution:

We will use simple unitary method to solve this question.

When A and B work together, they will complete 4*(1/8 + 1/12) = 5/6 work

Thus the remaining work = 1 - 5/6 = 1/6

Since C completes the entire work in 6 days, it will take him one day to finish 1/6 of work.