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's assume Peter had x letters at first.
After mailing out 4/5 of his letters, he was left with 1/5 * x letters.
He then received 30 new letters, so he had 1/5 * x + 30 letters.
After mailing out 1/6 of his letters, he was left with 5/6 * (1/5 * x + 30) letters.
He also received 15 new letters, so he had 5/6 * (1/5 * x + 30) + 15 letters.
According to the given information, Peter had 70 letters at the end. Therefore, we can create the equation:
5/6 * (1/5 * x + 30) + 15 = 70
Multiplying both sides of the equation by 6/5 to eliminate the denominator:
(1/5 * x + 30) + 18 = 84
1/5 * x + 48 = 84
1/5 * x = 36
Multiplying both sides of the equation by 5 to solve for x:
x = 36 * 5
x = <<36*5=180>>180
Therefore, Peter had 180 letters at first. Answer: \boxed{180}.
Let's assume that Peter initially had "x" letters.
After mailing out 4/5 of his letters, Peter had (4/5)x letters left. Therefore, (4/5)x + 30 = number of letters he had before mailing out 1/6 of his letters.
After mailing out 1/6 of his letters, Peter had (5/6)((4/5)x + 30) letters left. Therefore, (5/6)((4/5)x + 30) + 15 = 70 letters.
Simplifying the equation:
(5/6)((4/5)x + 30) + 15 = 70
Multiply both sides by 6/5:
(4/5)x + 30 + (6/5)(15) = 70
Multiply both sides by 5:
4x + 150 + 6(15) = 350
4x + 150 + 90 = 350
4x + 240 = 350
Subtract 240 from both sides:
4x = 110
Divide both sides by 4:
x = 27.5
Since we can't have a fraction of a letter, the initial number of letters, x, must be a whole number.
Therefore, Peter initially had 27.5 letters.
However, this is not a valid answer. It seems there is an error or missing information in the problem. Please double-check the problem statement.