# Q. What is the number of fives used in numbering a 260-page book?

a) 55

b) 56

c) 57

d) 60

Correct answer: b) 56

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

**To determine how many times the digit 5 appears in the page numbers of a 260-page book, we can analyze each digit place (units, tens, and hundreds) separately.**

## 1. Units Place:

Every 10 numbers, the units digit cycles from 0 to 9.

Therefore, the digit 5 appears once in every set of 10 numbers.

Total sets of 10 in 260 pages: 260/10 = 26 sets.

Occurrences in units place: 26 × 1 = 26

### 2. Tens Place:

The digit 5 appears in the tens place for every number from 50-59, 150-159, and 250-259.

Each of these ranges contains 10 numbers where the tens digit is 5.

Total ranges with 5 in tens place: 3 (i.e., 50-59, 150-159, 250-259)

Occurrences in tens place: 3 × 10 = 30

### 3. Hundreds Place:

Since the book has only 260 pages, the hundreds digit can be 0, 1, or 2.

The digit 5 does not appear in the hundreds place within this range.

Occurrences in hundreds place: 0

**Total Number of Fives:**

Units place: 26

Tens place: 30

Hundreds place: 0

Total: 26 + 30 + 0 = 56

Therefore, the correct answer is b) 56.