Peter the postman was managing some letters. After mailing out 4/5 of his letters, he received 30 new ones. He then mailed out 1/6 of his letters and received 15 more letters. In the end, Peter had 70 letters.
How many letters did he have at first?

Let original number of letters be L
After mailing out  and then receiving 30 new, he then had  

After mailing out  of . he then had  

Since after receiving 15 more, the final count amounted to 70, we then get: 
                                                Original number of letters, or L = 180

In order for the solution be clear, I present the given part in steps.

    1.  First, he mailed out 4/5 of the letters.

    2.  Then 30 new letters arrived.

    3.  Then mailed out 1/6 of his letters.

    4.  Then 15 more letters arrived.

    5.  At the end, were there 70 letters.


Solving backward, we see that immediately before step 4, were there 70-15 = 55 letters.


These 55 letters comprised 5/6 of what he had immediately before step 3.
Hence, 1/6 comprised 55/5 = 11 letter;  hence, immediately before step 3, there were 11*6 = 66 letters.


Hence, immediately before step 2, there were 66 - 30 = 36 letters.


These 36 letters comprised 1/5 of what he had at first.
Hence, at first, there there 36*5 = 180 letters.