A book contains 85 pages. A page is chosen at random. What is the probability that the sum of the digits on the page is 8?

Answer

To find the probability that the sum of the digits on a randomly chosen page from a book containing 85 pages is 8, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Pages:**
The book contains a total of 85 pages.

2. **Determine the Favorable Outcomes:**
We need to find the page numbers where the sum of the digits equals 8. Let's analyze the page numbers:

- For 1-digit pages (1 to 9):
- The only page is **8** (since 8 = 8).

- For 2-digit pages (10 to 85):
- **17** (1 + 7 = 8)
- **26** (2 + 6 = 8)
- **35** (3 + 5 = 8)
- **44** (4 + 4 = 8)
- **53** (5 + 3 = 8)
- **62** (6 + 2 = 8)
- **71** (7 + 1 = 8)
- **80** (8 + 0 = 8)

So, the favorable cases where the sum of the digits equals 8 are:
- 8
- 17
- 26
- 35
- 44
- 53
- 62
- 71
- 80

This gives us a total of **9 favorable outcomes**.

3. **Calculate the Probability:**
The probability \( P \) is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Here, the number of favorable outcomes is 9, and the total number of outcomes (total pages) is 85. Thus, we have:
\[ P = \frac{9}{85} \]

### Final Answer:
The probability that the sum of the digits on the page is 8 is \( \frac{9}{85} \).

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